f(x)=(e^x)/(x^2)

{ "query": { "display": "$$f\\left(x\\right)=\\frac{e^{x}}{x^{2}}$$", "symbolab_question": "FUNCTION#f(x)=\\frac{e^{x}}{x^{2}}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "Combination", "default": "\\mathrm{Domain}: x<0\\lor x>0<br/>\\mathrm{Range}: f(x)>0<br/>\\mathrm{Intercepts}: \\mathrm{None}<br/>\\mathrm{Asymptotes}: \\mathrm{Vertical}\\:x=0,\\mathrm{Horizontal}\\:y=0<br/>\\mathrm{Extreme\\:Points}: \\mathrm{Minimum}(2,\\frac{e^{2}}{4})", "interval": "\\mathrm{Domain}: (-\\infty ,0)\\cup (0,\\infty )<br/>\\mathrm{Range}: (0,\\infty )" }, "steps": { "type": "interim", "steps": [ { "type": "interim", "title": "Domain of $$\\frac{e^{x}}{x^{2}}\\::{\\quad}x<0\\lor\\:x>0$$", "steps": [ { "type": "definition", "title": "Domain definition", "text": "The domain of a function is the set of input or argument values for which the function is real and defined" }, { "type": "interim", "title": "Find undefined (singularity) points:$${\\quad}x=0$$", "input": "\\frac{e^{x}}{x^{2}}", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{e^{x}}{x^{2}}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$x^{2}=0:{\\quad}x=0$$", "input": "x^{2}=0", "steps": [ { "type": "step", "primary": "Apply rule $$x^n=0\\quad\\Rightarrow\\quad\\:x=0$$" }, { "type": "step", "result": "x=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The following points are undefined", "result": "x=0" } ], "meta": { "interimType": "Undefined Points 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6lOJ8p6oHaoHZ/UO5mSHLD1djVTVNfTJtGPIDKm6IXF/gNghAZYORfq9JtKwOORytPfV2oqc+75VRNLwQrPQk9pugt1g9hg3EUIob9qhnfZUgQUxJPyUNnGfVirkcwpVO92jAB5RE7BOdqi256y9oUI/L/NWLiz7WlXYJyi0meZnYoDSQN4diQ31DYw8UMxnuQ==" } }, { "type": "step", "primary": "The function domain", "result": "x<0\\lor\\:x>0" } ], "meta": { "solvingClass": "Function Domain", "interimType": "Function Domain Top 1Eq" } }, { "type": "interim", "title": "Range of $$\\frac{e^{x}}{x^{2}}:{\\quad}f\\left(x\\right)>0$$", "steps": [ { "type": "definition", "title": "Function range definition", "text": "The set of values of the dependent variable for which a function is defined" }, { "type": "step", "primary": "Find the minimum and maximum value in each defined interval and unite the results" }, { "type": "interim", "title": "Domain of $$\\frac{e^{x}}{x^{2}}\\::{\\quad}x<0\\lor\\:x>0$$", "steps": [ { "type": "definition", "title": "Domain definition", "text": "The domain of a function is the set of input or argument values for which the function is real and defined" }, { "type": "interim", "title": "Find undefined (singularity) points:$${\\quad}x=0$$", "input": "\\frac{e^{x}}{x^{2}}", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{e^{x}}{x^{2}}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$x^{2}=0:{\\quad}x=0$$", "input": "x^{2}=0", "steps": [ { "type": "step", "primary": "Apply rule $$x^n=0\\quad\\Rightarrow\\quad\\:x=0$$" }, { "type": "step", "result": "x=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The following points are undefined", "result": "x=0" } ], "meta": { "interimType": "Undefined Points 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6lOJ8p6oHaoHZ/UO5mSHLD1djVTVNfTJtGPIDKm6IXF/gNghAZYORfq9JtKwOORytPfV2oqc+75VRNLwQrPQk9pugt1g9hg3EUIob9qhnfZUgQUxJPyUNnGfVirkcwpVO92jAB5RE7BOdqi256y9oUI/L/NWLiz7WlXYJyi0meZnYoDSQN4diQ31DYw8UMxnuQ==" } }, { "type": "step", "primary": "The function domain", "result": "x<0\\lor\\:x>0" } ], "meta": { "solvingClass": "Function Domain", "interimType": "Function Domain Top 1Eq" } }, { "type": "interim", "title": "Extreme Points of $$\\frac{e^{x}}{x^{2}}:{\\quad}$$Minimum$$\\left(2,\\:\\frac{e^{2}}{4}\\right)$$", "steps": [ { "type": "definition", "title": "First Derivative Test definition", "text": "Suppose that $$x=c$$ is a critical point of $$f\\left(x\\right)$$ then, <br/>If $$f\\:{^{\\prime}}\\left(x\\right)>0$$ to the left of $$x=c$$ and $$f\\:{^{\\prime}}\\left(x\\right)<0$$ to the right of $$x=c$$ then $$x=c$$ is a local maximum.<br/>If $$f\\:{^{\\prime}}\\left(x\\right)<0$$ to the left of $$x=c$$ and $$f\\:{^{\\prime}}\\left(x\\right)>\\:0$$ to the right of $$x=c$$ then $$x=c$$ is a local minimum.<br/>If $$f\\:{^{\\prime}}\\left(x\\right)$$ is the same sign on both sides of $$x=c$$ then $$x=c$$ is neither a local maximum nor a local minimum." }, { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(x\\right)=\\frac{e^{x}\\left(x-2\\right)}{x^{3}}$$", "input": "\\frac{d}{dx}\\left(\\frac{e^{x}}{x^{2}}\\right)", "steps": [ { "type": "step", "primary": "Apply the Quotient Rule: $$\\left(\\frac{f}{g}\\right)'=\\frac{f'{\\cdot}g-g'{\\cdot}f}{g^{2}}$$", "result": "=\\frac{\\frac{d}{dx}\\left(e^{x}\\right)x^{2}-\\frac{d}{dx}\\left(x^{2}\\right)e^{x}}{\\left(x^{2}\\right)^{2}}" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(e^{x}\\right)=e^{x}$$", "input": "\\frac{d}{dx}\\left(e^{x}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(e^{x}\\right)=e^{x}$$", "result": "=e^{x}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjqxKAa6SkUhZrTPjmns35ik3hxk9aCfAWodBRxXgUexthpiW0WhiZGad41dobHknD/L0MoYg+CUn6oyL3EO7YobH0DM3/kWbWNIKRYne2WotxxYM+bRuETn+hrcvGmLsg==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkmb3s5xAUYje7fZWSRkdb2k3hxk9aCfAWodBRxXgUexcQsmN/cITrVSOMImEqe3fkeCBKuYKgaNJ253gLI69U7cjrVUqImvoUuRtb+2ccCzWsr9JoDNJaP7hueshcYJ6w==" } }, { "type": "step", "result": "=\\frac{e^{x}x^{2}-2xe^{x}}{\\left(x^{2}\\right)^{2}}" }, { "type": "interim", "title": "Simplify $$\\frac{e^{x}x^{2}-2xe^{x}}{\\left(x^{2}\\right)^{2}}:{\\quad}\\frac{e^{x}\\left(x-2\\right)}{x^{3}}$$", "input": "\\frac{e^{x}x^{2}-2xe^{x}}{\\left(x^{2}\\right)^{2}}", "result": "=\\frac{e^{x}\\left(x-2\\right)}{x^{3}}", "steps": [ { "type": "interim", "title": "Factor $$e^{x}x^{2}-2xe^{x}:{\\quad}xe^{x}\\left(x-2\\right)$$", "input": "e^{x}x^{2}-2xe^{x}", "result": "=\\frac{xe^{x}\\left(x-2\\right)}{\\left(x^{2}\\right)^{2}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=e^{x}xx-2xe^{x}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$xe^{x}$$", "result": "=xe^{x}\\left(x-2\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Simplify $$\\left(x^{2}\\right)^{2}:{\\quad}x^{4}$$", "input": "\\left(x^{2}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=x^{2\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=x^{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jd2F9SNQ4l8Y5LjObCinw96GQqufR6tr2vPxOUv7H+9UXCH/RczT7MepT8JUGxNwZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz13rvDT7XPh5g1GffZc9C9/" } }, { "type": "step", "result": "=\\frac{e^{x}x\\left(x-2\\right)}{x^{4}}" }, { "type": "step", "primary": "Cancel the common factor: $$x$$", "result": "=\\frac{e^{x}\\left(x-2\\right)}{x^{3}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GCORLVVHoNUKKIKCG8MW2BGOPVVNpdDH/kxsTI2YUVZ1q76C8OazfU2FcovI65TQA585Wz2Y8ioMtXlAhbC3eUntoSNn7Xag8VSkCd+TyFh3A+MYmDLVMW8ajGkszMT/ZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz15vgWPvL2Q7OhNZveWk3xcEY49VU2l0Mf+TGxMjZhRVkddNPdHlszcwQPnreyfM5M=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Find intervals:$${\\quad}$$Increasing$$:-\\infty\\:<x<0,\\:$$Decreasing$$:0<x<2,\\:$$Increasing$$:2<x<\\infty\\:$$", "input": "f\\:{^{\\prime}}\\left(x\\right)=\\frac{e^{x}\\left(x-2\\right)}{x^{3}}", "steps": [ { "type": "interim", "title": "Find the critical points:$${\\quad}x=0,\\:x=2$$", "steps": [ { "type": "definition", "title": "Critical point definition", "text": "Critical points are points where the function is defined and its derivative is zero or undefined" }, { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(x\\right)=0:{\\quad}x=2$$", "input": "\\frac{e^{x}\\left(x-2\\right)}{x^{3}}=0", "steps": [ { "type": "step", "primary": "$$\\frac{f\\left(x\\right)}{g\\left(x\\right)}=0{\\quad\\Rightarrow\\quad}f\\left(x\\right)=0$$", "result": "e^{x}\\left(x-2\\right)=0" }, { "type": "step", "primary": "Using the Zero Factor Principle:$$\\quad$$ If $$ab=0\\:$$then $$a=0\\:$$or $$b=0$$", "result": "e^{x}=0\\lor\\:x-2=0" }, { "type": "interim", "title": "Solve $$e^{x}=0:{\\quad}$$No Solution for $$x\\in\\mathbb{R}$$", "input": "e^{x}=0", "steps": [ { "type": "step", "primary": "$$a^{f\\left(x\\right)}$$ cannot be zero or negative for $$x\\in\\mathbb{R}$$", "result": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$x-2=0:{\\quad}x=2$$", "input": "x-2=0", "steps": [ { "type": "interim", "title": "Move $$2\\:$$to the right side", "input": "x-2=0", "result": "x=2", "steps": [ { "type": "step", "primary": "Add $$2$$ to both sides", "result": "x-2+2=0+2" }, { "type": "step", "primary": "Simplify", "result": "x=2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Verify Solutions:$${\\quad}x=2\\:$$True", "steps": [ { "type": "step", "primary": "Check the solutions by plugging them into $$\\frac{e^{x}\\left(x-2\\right)}{x^{3}}=0$$<br/>Remove the ones that don't agree with the equation." }, { "type": "interim", "title": "Plug in $$x=2:{\\quad}$$True", "input": "\\frac{e^{2}\\left(2-2\\right)}{2^{3}}=0", "steps": [ { "type": "interim", "title": "$$\\frac{e^{2}\\left(2-2\\right)}{2^{3}}=0$$", "input": "\\frac{e^{2}\\left(2-2\\right)}{2^{3}}", "steps": [ { "type": "interim", "title": "$$e^{2}\\left(2-2\\right)=0$$", "input": "e^{2}\\left(2-2\\right)", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$2-2=0$$", "result": "=0\\cdot\\:e^{2}" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7PpgbiFiym/DqgoHOnylETiAn9lkDfZkicUGkO3EF+Ip/suoH0JzK9gJYOwax5fT27/PB1ep1urYBGc+jjZ0+fg1F5/24c34qDescA26tcc0=" } }, { "type": "step", "result": "=\\frac{0}{2^{3}}" }, { "type": "step", "primary": "Apply rule $$\\frac{0}{a}=0,\\:a\\ne\\:0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WFXrvSSXKcE1RreO2RcH3SP8IK1kjM00DV1G7E+y5QPdd47a0hQ8flDbGsI5To1dTbAOxT8wOTlsw5gGf+Hdr1NbbqpyK7JQEZdATEJR51iCVEYzSCtRx3J1efhO1qcD3IXjn3aOz/DkLLPMfmW3tg==" } }, { "type": "step", "primary": "$$0=0$$" }, { "type": "step", "result": "\\mathrm{True}" } ], "meta": { "interimType": "Generic Plug 1Eq" } } ], "meta": { "interimType": "Check Solutions Plug Preface 1Eq" } }, { "type": "step", "primary": "The solution is", "result": "x=2" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "interim", "title": "Find undefined (singularity) points of $$f\\:{^{\\prime}}\\left(x\\right):{\\quad}x=0$$", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{e^{x}\\left(x-2\\right)}{x^{3}}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$x^{3}=0:{\\quad}x=0$$", "input": "x^{3}=0", "steps": [ { "type": "step", "primary": "Apply rule $$x^n=0\\quad\\Rightarrow\\quad\\:x=0$$" }, { "type": "step", "result": "x=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The following points are undefined", "result": "x=0" } ], "meta": { "interimType": "Undefined Points 0Eq" } }, { "type": "step", "result": "x=0,\\:x=2" } ], "meta": { "interimType": "Explore Function Slope Zero Title 0Eq" } }, { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(x\\right)>0:{\\quad}x<0\\lor\\:x>2$$", "input": "\\frac{e^{x}\\left(x-2\\right)}{x^{3}}>0", "steps": [ { "type": "interim", "title": "Identify the intervals", "result": "x<0\\lor\\:x>2", "steps": [ { "type": "interim", "title": "Find the signs of the factors of $$\\frac{e^{x}\\left(x-2\\right)}{x^{3}}$$", "input": "\\frac{e^{x}\\left(x-2\\right)}{x^{3}}", "steps": [ { "type": "interim", "title": "Find the signs of $$e^{x}:{\\quad}$$Positive for all real values", "input": "e^{x}>0", "steps": [ { "type": "step", "primary": "If $$a>0,\\:a^{f\\left(x\\right)}\\:$$is greater than 0", "secondary": [ "$$a=e$$" ], "result": "\\mathrm{True\\:for\\:all}\\:x\\in\\mathbb{R}" } ], "meta": { "interimType": "Find Sign 1Eq" } }, { "type": "interim", "title": "Find the signs of $$x-2$$", "steps": [ { "type": "interim", "title": "$$x-2=0:{\\quad}x=2$$", "input": "x-2=0", "steps": [ { "type": "interim", "title": "Move $$2\\:$$to the right side", "input": "x-2=0", "result": "x=2", "steps": [ { "type": "step", "primary": "Add $$2$$ to both sides", "result": "x-2+2=0+2" }, { "type": "step", "primary": "Simplify", "result": "x=2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "interim", "title": "$$x-2<0:{\\quad}x<2$$", "input": "x-2<0", "steps": [ { "type": "interim", "title": "Move $$2\\:$$to the right side", "input": "x-2<0", "result": "x<2", "steps": [ { "type": "step", "primary": "Add $$2$$ to both sides", "result": "x-2+2<0+2" }, { "type": "step", "primary": "Simplify", "result": "x<2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], 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1Eq" } }, { "type": "interim", "title": "Combine $$x=2\\:$$ with domain:$${\\quad}x=2$$", "input": "x=2\\land\\:\\left(x<0\\lor\\:x>0\\right)", "steps": [ { "type": "step", "primary": "Simplify", "result": "x=2" } ], "meta": { "interimType": "Combine With Domain 1Eq" } }, { "type": "interim", "title": "Combine $$-\\infty\\:<x<0\\:$$ with domain:$${\\quad}-\\infty\\:<x<0$$", "input": "-\\infty\\:<x<0\\land\\:\\left(x<0\\lor\\:x>0\\right)", "steps": [ { "type": "step", "primary": "Simplify", "result": "-\\infty\\:<x<0" } ], "meta": { "interimType": "Combine With Domain 1Eq" } }, { "type": "interim", "title": "Combine $$2<x<\\infty\\:\\:$$ with domain:$${\\quad}2<x<\\infty\\:$$", "input": "2<x<\\infty\\:\\land\\:\\left(x<0\\lor\\:x>0\\right)", "steps": [ { "type": "step", "primary": "Simplify", "result": "2<x<\\infty\\:" } ], "meta": { "interimType": "Combine With Domain 1Eq" } }, { "type": "interim", "title": "Combine $$0<x<2\\:$$ with domain:$${\\quad}0<x<2$$", "input": "0<x<2\\land\\:\\left(x<0\\lor\\:x>0\\right)", "steps": [ { "type": "step", "primary": "Simplify", "result": "0<x<2" } ], "meta": { "interimType": "Combine With Domain 1Eq" } }, { "type": "step", "result": "-\\infty\\:<x<0,\\:0<x<2,\\:x=2,\\:2<x<\\infty\\:" } ], "meta": { "interimType": "Combine Intervals With Domain 0Eq" } }, { "type": "step", "primary": "Summary of the monotone intervals behavior", "secondary": [ "$$\\begin{array}{|c|c|c|c|c|}\\hline &-\\infty <x<0&0<x<2&x=2&2<x<\\infty \\\\\\hline \\mathrm{Sign}&f {^{\\prime}}(x)>0&f {^{\\prime}}(x)<0&f {^{\\prime}}(x)=0&f {^{\\prime}}(x)>0\\\\\\hline \\mathrm{Behavior}&\\mathrm{Increasing}&\\mathrm{Decreasing}&\\mathrm{Minimum}&\\mathrm{Increasing}\\\\\\hline \\end{array}$$" ] }, { "type": "step", "result": "\\mathrm{Increasing}:-\\infty\\:<x<0,\\:\\mathrm{Decreasing}:0<x<2,\\:\\mathrm{Increasing}:2<x<\\infty\\:" } ], "meta": { "interimType": "Function Find Intervals 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGUL8PhNw/UnccDjDvkOKjBsievSRG8FbtJ67MZcqchnBIfFLvhIw/699NIWZ6gDQnSGLMy6w719XZab5nwoBBElSIPF8OtyNv5ccXtMPbG7frRVnYC9rQrqhhG8PfLUSo3oe/oyhMy2+1TQhDBd2fyD9rFOlVgJh7dgabqjdJ3AnM3fsgDv9oMH72SfJx2Nf" } }, { "type": "interim", "title": "Plug $$x=2\\:$$into $$\\frac{e^{x}}{x^{2}}:{\\quad}\\frac{e^{2}}{4}$$", "input": "\\frac{e^{2}}{2^{2}}", "result": "\\mathrm{Minimum}\\left(2,\\:\\frac{e^{2}}{4}\\right)", "steps": [ { "type": "step", "primary": "Simplify", "result": "\\frac{e^{2}}{4}" } ], "meta": { "interimType": "Generic Plug Into Specific 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3d3CfN4XlvQgt3kPeYLw7cVZFeBK9phrhgsp3f4G7xsqOVy+4uzYex88fKU9o5iIJkS3dlcCKpQTQcheuut7MkS4X77HvSt7slSZWQ9ahxI/TQzK6OUCL7blIKt1RiPGFTifKeqB2qB2f1DuZkhyw9pXXVBzYmEcbFxkcAsY6b8g==" } } ], "meta": { "solvingClass": "Function Extreme", "interimType": "Extreme Points Table Top 1Eq" } }, { "type": "interim", "title": "Find the range for the interval $$-\\infty\\:<x<0:{\\quad}0<f\\left(x\\right)<\\infty\\:$$", "steps": [ { "type": "step", "primary": "Compute the values of the function at the edges of the interval:" }, { "type": "interim", "title": "$$\\lim_{x\\to\\:-\\infty\\:}\\left(\\frac{e^{x}}{x^{2}}\\right)=0$$", "input": "\\lim_{x\\to\\:-\\infty\\:}\\left(\\frac{e^{x}}{x^{2}}\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}[\\frac{f\\left(x\\right)}{g\\left(x\\right)}]=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:\\quad\\lim_{x\\to{a}}{g\\left(x\\right)}\\neq0$$<br/>With the exception of indeterminate form", "result": "=\\frac{\\lim_{x\\to\\:-\\infty\\:}\\left(e^{x}\\right)}{\\lim_{x\\to\\:-\\infty\\:}\\left(x^{2}\\right)}", "meta": { "title": { "extension": "Indeterminate Forms:<br/>$$\\frac{\\pm\\infty}{\\pm\\infty}$$<br/>$$\\frac{0}{0}$$<br/>$$\\pm\\infty\\cdot0$$<br/>$$0^0$$<br/>$$1^{\\pm\\infty}$$<br/>$$\\infty^{0}$$<br/>$$\\infty-\\infty$$" } } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:-\\infty\\:}\\left(e^{x}\\right)=0$$", "input": "\\lim_{x\\to\\:-\\infty\\:}\\left(e^{x}\\right)", "steps": [ { "type": "step", "primary": "Apply the common limit: $$\\lim_{x\\to\\:-\\infty\\:}\\left(e^{x}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:-\\infty\\:}\\left(x^{2}\\right)=\\infty\\:$$", "input": "\\lim_{x\\to\\:-\\infty\\:}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$\\lim_{x\\to\\pm\\infty}\\left(ax^{n}+\\cdots+bx+c\\right)=\\infty,\\:a>0,\\:$$n is even<br/>$$a=1,\\:n=2$$", "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+jbUbsVfpMbXIJ0UHfO5sW8yQ0stCqMVssrXgR2l7RB2D7i9sYpm+RqCkMjInr+m9btD533cgzHO58Vzwl73PAudFfSb5EGya/un1pgRn2a0wKciUtQg/2njDjXZOJLx5pYbHEl1zYMbNoc/CxPb8Tmqm40qzyGVY+WsRGEnpKJarGDlHNgwSdrYmV8S+RDA+D9a+DqAkCA8aT0cL1QXCtbEntzkBfjeX+WSvPaeRIo=" } }, { "type": "step", "result": "=\\frac{0}{\\infty\\:}" }, { "type": "step", "primary": "Apply Infinity Property: $$\\frac{0}{\\infty\\:}=0$$", "result": "=0", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:0-}\\left(\\frac{e^{x}}{x^{2}}\\right)=\\infty\\:$$", "input": "\\lim_{x\\to\\:0-}\\left(\\frac{e^{x}}{x^{2}}\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{e^{x}}{x^{2}}=e^{x}\\frac{1}{x^{2}}$$", "input": "\\frac{e^{x}}{x^{2}}", "steps": [ { "type": "step", "primary": "Rewrite $$e^{x}$$ as $$e^{x}\\cdot\\:1$$", "result": "=\\frac{e^{x}\\cdot\\:1}{x^{2}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a*b}{c}=a*\\frac{b}{c}$$", "secondary": [ "$$\\frac{e^{x}\\cdot\\:1}{x^{2}}=e^{x}\\frac{1}{x^{2}}$$" ], "result": "=e^{x}\\frac{1}{x^{2}}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7MPPA0PsU1dpF0ZPHOwf2N9SUzgALSYbl/aGC/IIM1p+rju+5Z51e/ZZSD3gRHwjBb99ixNf8xygcQ1JySdWUentN0aAmwaJ0/Zz0jfC2PcTxi2FAvU9I2PcaucfrzxFSQb2CQJd2BC0nM53oKErhOgD+MHlEtY8j9TbXGUSRhH4WRUPDnIQXYMTDiaxDMpN7" } }, { "type": "step", "result": "=\\lim_{x\\to\\:0-}\\left(e^{x}\\frac{1}{x^{2}}\\right)" }, { "type": "step", "primary": "$$\\lim_{x\\to{a}}[f\\left(x\\right)\\cdot{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\cdot\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>With the exception of indeterminate form", "result": "=\\lim_{x\\to\\:0-}\\left(e^{x}\\right)\\cdot\\:\\lim_{x\\to\\:0-}\\left(\\frac{1}{x^{2}}\\right)", "meta": { "title": { "extension": "Indeterminate Forms:<br/>$$\\frac{\\pm\\infty}{\\pm\\infty}$$<br/>$$\\frac{0}{0}$$<br/>$$\\pm\\infty\\cdot0$$<br/>$$0^0$$<br/>$$1^{\\pm\\infty}$$<br/>$$\\infty^{0}$$<br/>$$\\infty-\\infty$$" } } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:0-}\\left(e^{x}\\right)=1$$", "input": "\\lim_{x\\to\\:0-}\\left(e^{x}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=0$$", "result": "=e^{0}", "meta": { "title": { "extension": "Limit properties - if the limit of f(x), and g(x) exists, then:<br/>$$\\bullet\\quad\\lim_{x\\to\\:a}\\left(x\\right)=a$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[\\left(f\\left(x\\right)\\right)^c]=\\left(\\lim_{x\\to{a}}{f\\left(x\\right)}\\right)^c$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\pm{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\pm\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\cdot{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\cdot\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:$$where $$\\lim_{x\\to{a}}g\\left(x\\right)\\neq0$$" } } }, { "type": "step", "primary": "Simplify", "result": "=1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:0-}\\left(\\frac{1}{x^{2}}\\right)=\\infty\\:$$", "input": "\\lim_{x\\to\\:0-}\\left(\\frac{1}{x^{2}}\\right)", "steps": [ { "type": "step", "primary": "For $$x\\:$$approaching $$0\\:$$from the left$$,\\:x<0\\quad\\Rightarrow\\quad\\:x^{2}>0$$", "secondary": [ "The denominator is a positive quantity approaching 0 from the right" ], "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=1\\cdot\\:\\infty\\:" }, { "type": "step", "primary": "Apply Infinity Property: $$c\\cdot\\infty=\\infty$$", "result": "=\\infty\\:", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "primary": "Minimum function value at the domain interval $$-\\infty\\:<x<0$$ is $$0$$" }, { "type": "step", "primary": "Maximum function value at the domain interval $$-\\infty\\:<x<0$$ is $$\\infty\\:$$" }, { "type": "step", "primary": "Therefore the range of $$\\frac{e^{x}}{x^{2}}\\:$$ at the domain interval $$-\\infty\\:<x<0\\:$$ is" }, { "type": "step", "result": "0<f\\left(x\\right)<\\infty\\:" } ], "meta": { "interimType": "Function Range Interval 1Eq" } }, { "type": "interim", "title": "Find the range for the interval $$0<x<\\infty\\::{\\quad}\\frac{e^{2}}{4}\\le\\:f\\left(x\\right)<\\infty\\:$$", "steps": [ { "type": "step", "primary": "Compute the values of the function at the edges of the interval:" }, { "type": "interim", "title": "$$\\lim_{x\\to\\:0+}\\left(\\frac{e^{x}}{x^{2}}\\right)=\\infty\\:$$", "input": "\\lim_{x\\to\\:0+}\\left(\\frac{e^{x}}{x^{2}}\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{e^{x}}{x^{2}}=e^{x}\\frac{1}{x^{2}}$$", "input": "\\frac{e^{x}}{x^{2}}", "steps": [ { "type": "step", "primary": "Rewrite $$e^{x}$$ as $$e^{x}\\cdot\\:1$$", "result": "=\\frac{e^{x}\\cdot\\:1}{x^{2}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a*b}{c}=a*\\frac{b}{c}$$", "secondary": [ "$$\\frac{e^{x}\\cdot\\:1}{x^{2}}=e^{x}\\frac{1}{x^{2}}$$" ], "result": "=e^{x}\\frac{1}{x^{2}}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7MPPA0PsU1dpF0ZPHOwf2N9SUzgALSYbl/aGC/IIM1p+rju+5Z51e/ZZSD3gRHwjBb99ixNf8xygcQ1JySdWUentN0aAmwaJ0/Zz0jfC2PcTxi2FAvU9I2PcaucfrzxFSQb2CQJd2BC0nM53oKErhOgD+MHlEtY8j9TbXGUSRhH4WRUPDnIQXYMTDiaxDMpN7" } }, { "type": "step", "result": "=\\lim_{x\\to\\:0+}\\left(e^{x}\\frac{1}{x^{2}}\\right)" }, { "type": "step", "primary": "$$\\lim_{x\\to{a}}[f\\left(x\\right)\\cdot{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\cdot\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>With the exception of indeterminate form", "result": "=\\lim_{x\\to\\:0+}\\left(e^{x}\\right)\\cdot\\:\\lim_{x\\to\\:0+}\\left(\\frac{1}{x^{2}}\\right)", "meta": { "title": { "extension": "Indeterminate Forms:<br/>$$\\frac{\\pm\\infty}{\\pm\\infty}$$<br/>$$\\frac{0}{0}$$<br/>$$\\pm\\infty\\cdot0$$<br/>$$0^0$$<br/>$$1^{\\pm\\infty}$$<br/>$$\\infty^{0}$$<br/>$$\\infty-\\infty$$" } } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:0+}\\left(e^{x}\\right)=1$$", "input": "\\lim_{x\\to\\:0+}\\left(e^{x}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=0$$", "result": "=e^{0}", "meta": { "title": { "extension": "Limit properties - if the limit of f(x), and g(x) exists, then:<br/>$$\\bullet\\quad\\lim_{x\\to\\:a}\\left(x\\right)=a$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[\\left(f\\left(x\\right)\\right)^c]=\\left(\\lim_{x\\to{a}}{f\\left(x\\right)}\\right)^c$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\pm{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\pm\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\cdot{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\cdot\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:$$where $$\\lim_{x\\to{a}}g\\left(x\\right)\\neq0$$" } } }, { "type": "step", "primary": "Simplify", "result": "=1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:0+}\\left(\\frac{1}{x^{2}}\\right)=\\infty\\:$$", "input": "\\lim_{x\\to\\:0+}\\left(\\frac{1}{x^{2}}\\right)", "steps": [ { "type": "step", "primary": "For $$x\\:$$approaching $$0\\:$$from the right$$,\\:x>0\\quad\\Rightarrow\\quad\\:x^{2}>0$$", "secondary": [ "The denominator is a positive quantity approaching 0 from the right" ], "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=1\\cdot\\:\\infty\\:" }, { "type": "step", "primary": "Apply Infinity Property: $$c\\cdot\\infty=\\infty$$", "result": "=\\infty\\:", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{e^{x}}{x^{2}}\\right)=\\infty\\:$$", "input": "\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{e^{x}}{x^{2}}\\right)", "steps": [ { "type": "interim", "title": "Apply L'Hopital's Rule", "input": "\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{e^{x}}{x^{2}}\\right)", "result": "=\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{e^{x}}{2x}\\right)", "steps": [ { "type": "definition", "title": "L'Hopital Theorem:", "text": "For $$\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right),\\:\\mathrm{if}\\:\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{0}{0}\\quad\\mathrm{or}\\quad\\lim_{x\\to\\:a}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{\\pm\\infty}{\\pm\\infty},\\:\\mathrm{then}$$<br/>$$\\quad\\quad\\quad\\bold{\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\lim_{x\\to{a}}\\left(\\frac{f^{'}\\left(x\\right)}{g^{'}\\left(x\\right)}\\right)}$$" }, { "type": "interim", "title": "Test L'Hopital condition: $$\\frac{\\infty\\:}{\\infty\\:}$$", "result": "=\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{\\left(e^{x}\\right)^{^{\\prime}}}{\\left(x^{2}\\right)^{^{\\prime}}}\\right)", "steps": [ { "type": "interim", "title": "$$\\lim_{x\\to\\:\\infty\\:}\\left(e^{x}\\right)=\\infty\\:$$", "input": "\\lim_{x\\to\\:\\infty\\:}\\left(e^{x}\\right)", "steps": [ { "type": "step", "primary": "Apply the common limit: $$\\lim_{x\\to\\:\\infty\\:}\\left(e^{x}\\right)=\\infty\\:$$", "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:\\infty\\:}\\left(x^{2}\\right)=\\infty\\:$$", "input": "\\lim_{x\\to\\:\\infty\\:}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$\\lim_{x\\to\\pm\\infty}\\left(ax^{n}+\\cdots+bx+c\\right)=\\infty,\\:a>0,\\:$$n is even<br/>$$a=1,\\:n=2$$", "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+jbUbsVfpMbXIJ0UHfO5saL0uDZ6PHhP1GqJznWyHwkaMS1C7LHR3Ox3wzQnf9x+cAgzx9zrYTFpwIg0trnpZTVTxY0hilST8p7oi8GAvx7QF2UzOp5Bb4XpeTJ7DlfnwhWwQYyckKycWgIwD4f3sBi2Sg2N1jZXcumfy0+UpgCi9Lg2ejx4T9Rqic51sh8JeOKMC/3/n4eO0zNsI03rcdnUMsHFIy5Nsmxty17Bu2Q=" } }, { "type": "step", "primary": "Meets L'hopital condition of: $$\\frac{\\infty\\:}{\\infty\\:}$$", "result": "=\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{\\left(e^{x}\\right)^{^{^{\\prime}}}}{\\left(x^{2}\\right)^{^{^{\\prime}}}}\\right)" } ], "meta": { "interimType": "Lhopital Condition 1Eq" } }, { "type": "interim", "title": "$$\\left(e^{x}\\right)'=e^{x}$$", "input": "\\frac{d}{dx}\\left(e^{x}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(e^{x}\\right)=e^{x}$$", "result": "=e^{x}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjqxKAa6SkUhZrTPjmns35ik3hxk9aCfAWodBRxXgUexthpiW0WhiZGad41dobHknD/L0MoYg+CUn6oyL3EO7Yp1B2hMQUtdwgYJZ1r+cLQNJLd1ohke2Wgml78++2zI0g==" } }, { "type": "interim", "title": "$$\\left(x^{2}\\right)'=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkmb3s5xAUYje7fZWSRkdb2k3hxk9aCfAWodBRxXgUexcQsmN/cITrVSOMImEqe3fudHvcmjs8iJXuaQ+nxJX156Xz0OParmIowNHqIzDBN8" } }, { "type": "step", "result": "=\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{e^{x}}{2x}\\right)" } ], "meta": { "interimType": "Lhopital Theorem 0Eq", "practiceLink": "/practice/limits-practice#area=main&subtopic=L'Hopital%20Rule", "practiceTopic": "L'hopital Rule", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+jbUbsVfpMbXIJ0UHfO5saL0uDZ6PHhP1GqJznWyHwmWn2tyts4ihSIs6Ah0p2yXtcngnTW1AWFeb02vni0XLbmIKnaLGSq3SuzYRYyGsT2mmyLIEdPsejKo8OmKWvZOgSEHRda+G5BHM5FRE2/NG+7LUvr52HEm2rBNGzBdDyCk+q2xbw+ySI/eV65OSy9291sIFIsmZBinqaLjpKnVeU3kCh3oevUunZ7/b0qFKBSyIAv17r6SEQ+f7Q/d2FLEdYUDEApzDR4jzQUGsjov1w==" } }, { "type": "interim", "title": "Apply L'Hopital's Rule", "input": "\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{e^{x}}{2x}\\right)", "result": "=\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{e^{x}}{2}\\right)", "steps": [ { "type": "definition", "title": "L'Hopital Theorem:", "text": "For $$\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right),\\:\\mathrm{if}\\:\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{0}{0}\\quad\\mathrm{or}\\quad\\lim_{x\\to\\:a}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{\\pm\\infty}{\\pm\\infty},\\:\\mathrm{then}$$<br/>$$\\quad\\quad\\quad\\bold{\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\lim_{x\\to{a}}\\left(\\frac{f^{'}\\left(x\\right)}{g^{'}\\left(x\\right)}\\right)}$$" }, { "type": "interim", "title": "Test L'Hopital condition: $$\\frac{\\infty\\:}{\\infty\\:}$$", "result": "=\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{\\left(e^{x}\\right)^{^{\\prime}}}{\\left(2x\\right)^{^{\\prime}}}\\right)", "steps": [ { "type": "interim", "title": "$$\\lim_{x\\to\\:\\infty\\:}\\left(e^{x}\\right)=\\infty\\:$$", "input": "\\lim_{x\\to\\:\\infty\\:}\\left(e^{x}\\right)", "steps": [ { "type": "step", "primary": "Apply the common limit: $$\\lim_{x\\to\\:\\infty\\:}\\left(e^{x}\\right)=\\infty\\:$$", "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:\\infty\\:}\\left(2x\\right)=\\infty\\:$$", "input": "\\lim_{x\\to\\:\\infty\\:}\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$\\lim_{x\\to\\infty}\\left(ax^{n}+\\cdots+bx+c\\right)=\\infty,\\:a>0,\\:$$n is odd<br/>$$a=2,\\:n=1$$", "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+jbUbsVfpMbXIJ0UHfO5saL0uDZ6PHhP1GqJznWyHwlV+qg7Q+0fNarYODiCp208gWZR3zCAilVrsafPmsT5abA9Nb3zLvO1gWJufFEd2tIwFfVA2P3O5Qj7EDGNsqJdP3yb3suSQQSaGuJSfY5SMAaS5CGs21zxA42OGelXxJHeyE0iYvjxncDM4tBgVfPAyD8MhUKmOPCU9J4rK7X6Ug==" } }, { "type": "step", "primary": "Meets L'hopital condition of: $$\\frac{\\infty\\:}{\\infty\\:}$$", "result": "=\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{\\left(e^{x}\\right)^{^{^{\\prime}}}}{\\left(2x\\right)^{^{^{\\prime}}}}\\right)" } ], "meta": { "interimType": "Lhopital Condition 1Eq" } }, { "type": "interim", "title": "$$\\left(e^{x}\\right)'=e^{x}$$", "input": "\\frac{d}{dx}\\left(e^{x}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(e^{x}\\right)=e^{x}$$", "result": "=e^{x}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjqxKAa6SkUhZrTPjmns35ik3hxk9aCfAWodBRxXgUexthpiW0WhiZGad41dobHknD/L0MoYg+CUn6oyL3EO7Yp1B2hMQUtdwgYJZ1r+cLQNJLd1ohke2Wgml78++2zI0g==" } }, { "type": "interim", "title": "$$\\left(2x\\right)'=2$$", "input": "\\frac{d}{dx}\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYg2sQzwGEAAPyDk8n13Ps8XZGku9zFkxwe1dTH8vycb94wHsFp27x8BxzSfXYcuPliXKvlioccNR9e+UjAgV/3qwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{e^{x}}{2}\\right)" } ], "meta": { "interimType": "Lhopital Theorem 0Eq", "practiceLink": "/practice/limits-practice#area=main&subtopic=L'Hopital%20Rule", "practiceTopic": "L'hopital Rule", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+jbUbsVfpMbXIJ0UHfO5saL0uDZ6PHhP1GqJznWyHwmWn2tyts4ihSIs6Ah0p2yXChKL+0uARTKwznN1sBA+oljlDP5aX4SprmJ1raef0WDidrg8DC82xolA3cBpRMkxM13xa8ZsWq5c7JLPonGhI7OhsTL2emzAKqGHxCz6tu2b4cvchfV+5bNofnpxdk5a35owguej18ZHfjpsA1+YiAS4M5VpC8qh+oehjmM1qmwtqhtTKB1YevrvLbYub3J5BnQ/fS3TdY5cJWNofcUKJg==" } }, { "type": "step", "primary": "$$\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$", "result": "=\\frac{1}{2}\\cdot\\:\\lim_{x\\to\\:\\infty\\:}\\left(e^{x}\\right)" }, { "type": "step", "primary": "Apply the common limit: $$\\lim_{x\\to\\:\\infty\\:}\\left(e^{x}\\right)=\\infty\\:$$", "result": "=\\frac{1}{2}\\cdot\\:\\infty\\:" }, { "type": "step", "primary": "Apply Infinity Property: $$c\\cdot\\infty=\\infty$$", "result": "=\\infty\\:", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "primary": "The interval has a minimum point at $$x=2\\:$$ with value $$f\\left(2\\right)=\\frac{e^{2}}{4}$$" }, { "type": "step", "primary": "Combine the function value at the edge with the extreme points of the function in the interval:" }, { "type": "step", "primary": "Minimum function value at the domain interval $$0<x<\\infty\\:$$ is $$\\frac{e^{2}}{4}$$" }, { "type": "step", "primary": "Maximum function value at the domain interval $$0<x<\\infty\\:$$ is $$\\infty\\:$$" }, { "type": "step", "primary": "Therefore the range of $$\\frac{e^{x}}{x^{2}}\\:$$ at the domain interval $$0<x<\\infty\\:\\:$$ is" }, { "type": "step", "result": "\\frac{e^{2}}{4}\\le\\:f\\left(x\\right)<\\infty\\:" } ], "meta": { "interimType": "Function Range Interval 1Eq" } }, { "type": "step", "primary": "Combine the ranges of all domain intervals to obtain the function range ", "secondary": [ "$$f\\left(x\\right)>0\\lor\\:f\\left(x\\right)\\ge\\:\\frac{e^{2}}{4}$$" ], "result": "f\\left(x\\right)>0" } ], "meta": { "solvingClass": "Function Range", "interimType": "Function Range Top 1Eq" } }, { "type": "interim", "title": "Axis interception points of $$\\frac{e^{x}}{x^{2}}:\\quad\\:$$None", "steps": [ { "type": "interim", "title": "$$x-$$axis interception points of $$\\frac{e^{x}}{x^{2}}:{\\quad}$$None", "input": "\\frac{e^{x}}{x^{2}}", "steps": [ { "type": "definition", "title": "x-axis interception points definition", "text": "x-intercept is a point on the graph where $$y=0$$" }, { "type": "interim", "title": "Solve $$\\frac{e^{x}}{x^{2}}=0:{\\quad}$$No Solution for $$x\\in\\mathbb{R}$$", "input": "\\frac{e^{x}}{x^{2}}=0", "steps": [ { "type": "step", "primary": "$$\\frac{f\\left(x\\right)}{g\\left(x\\right)}=0{\\quad\\Rightarrow\\quad}f\\left(x\\right)=0$$", "result": "e^{x}=0" }, { "type": "step", "primary": "$$a^{f\\left(x\\right)}$$ cannot be zero or negative for $$x\\in\\mathbb{R}$$", "result": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "\\mathrm{No\\:x-axis\\:interception\\:points}" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Interception X Points Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMXsqdoP/+t8mkG9iMyF4x8bt5g+KB4e7b6i2FR+k9P7NTifKeqB2qB2f1DuZkhyw9Ytn3TyLDmu4LrVnxZ+jEEYfmT5KZV+W/Wb2p4ioO/4+O5FEYlptVV2+TVuQDnXvIYZtW9kSIaJvzqiZ7at7yuaOVy+4uzYex88fKU9o5iIKwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "$$y-$$axis interception point of $$\\frac{e^{x}}{x^{2}}:{\\quad}$$None", "input": "\\frac{e^{x}}{x^{2}}", "steps": [ { "type": "definition", "title": "y-axis interception points definition", "text": "$$y$$-intercept is the point on the graph where $$x=0$$" }, { "type": "interim", "title": "Domain of $$\\frac{e^{x}}{x^{2}}\\::{\\quad}x<0\\lor\\:x>0$$", "steps": [ { "type": "definition", "title": "Domain definition", "text": "The domain of a function is the set of input or argument values for which the function is real and defined" }, { "type": "interim", "title": "Find undefined (singularity) points:$${\\quad}x=0$$", "input": "\\frac{e^{x}}{x^{2}}", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{e^{x}}{x^{2}}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$x^{2}=0:{\\quad}x=0$$", "input": "x^{2}=0", "steps": [ { "type": "step", "primary": "Apply rule $$x^n=0\\quad\\Rightarrow\\quad\\:x=0$$" }, { "type": "step", "result": "x=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The following points are undefined", "result": "x=0" } ], "meta": { "interimType": "Undefined Points 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6lOJ8p6oHaoHZ/UO5mSHLD1djVTVNfTJtGPIDKm6IXF/gNghAZYORfq9JtKwOORytPfV2oqc+75VRNLwQrPQk9pugt1g9hg3EUIob9qhnfZUgQUxJPyUNnGfVirkcwpVO92jAB5RE7BOdqi256y9oUI/L/NWLiz7WlXYJyi0meZnYoDSQN4diQ31DYw8UMxnuQ==" } }, { "type": "step", "primary": "The function domain", "result": "x<0\\lor\\:x>0" } ], "meta": { "solvingClass": "Function Domain", "interimType": "Function Domain Top 1Eq" } }, { "type": "step", "primary": "Since $$x=0\\:$$is not in domain", "result": "\\mathrm{No\\:y-axis\\:interception\\:point}" }, { "type": "step", "result": "\\mathrm{None}" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Interception Y Points Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMoY5LPa3x5862ED2Fb21Kz+w/rOyd5aPv89U4pRBmcfmT+Ewt2VMLE3/oFmI5TlJ3fmghG98Q86OXAzp/EDJXoj4wTfPZsaLh75d82KZhfvQZ9T38S5pPc5mtR18r12ZvDxm42DL15aq9NMMpp1Iw7HV3bU20EXZfHq/ezpYWZrckt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "\\mathrm{None}" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Function Intercepts Top 2Eq" } }, { "type": "interim", "title": "Asymptotes of $$\\frac{e^{x}}{x^{2}}:\\quad\\:$$Vertical$$:\\:x=0,\\:$$Horizontal$$:\\:y=0$$", "steps": [ { "type": "interim", "title": "Vertical asymptotes of $$\\frac{e^{x}}{x^{2}}:{\\quad}x=0$$", "input": "\\frac{e^{x}}{x^{2}}", "steps": [ { "type": "step", "primary": "Go over every undefined point $$x=a$$ and check if at least one of the following statements is true:<br/>$${\\quad}\\lim_{x\\to{a^{-}}}f\\left(x\\right)=\\pm\\infty$$<br/>$${\\quad}\\lim_{x\\to{a^{+}}}f\\left(x\\right)=\\pm\\infty$$" }, { "type": "interim", "title": "Find undefined (singularity) points:$${\\quad}x=0$$", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{e^{x}}{x^{2}}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$x^{2}=0:{\\quad}x=0$$", "input": "x^{2}=0", "steps": [ { "type": "step", "primary": "Apply rule $$x^n=0\\quad\\Rightarrow\\quad\\:x=0$$" }, { "type": "step", "result": "x=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The following points are undefined", "result": "x=0" } ], "meta": { "interimType": "Undefined Points 0Eq" } }, { "type": "interim", "title": "Check for a vertical asymptote at $$x=0:{\\quad}$$True", "steps": [ { "type": "interim", "title": "$$\\lim_{x\\to\\:0-}\\left(\\frac{e^{x}}{x^{2}}\\right)=\\infty\\:$$", "input": "\\lim_{x\\to\\:0-}\\left(\\frac{e^{x}}{x^{2}}\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{e^{x}}{x^{2}}=e^{x}\\frac{1}{x^{2}}$$", "input": "\\frac{e^{x}}{x^{2}}", "steps": [ { "type": "step", "primary": "Rewrite $$e^{x}$$ as $$e^{x}\\cdot\\:1$$", "result": "=\\frac{e^{x}\\cdot\\:1}{x^{2}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a*b}{c}=a*\\frac{b}{c}$$", "secondary": [ "$$\\frac{e^{x}\\cdot\\:1}{x^{2}}=e^{x}\\frac{1}{x^{2}}$$" ], "result": "=e^{x}\\frac{1}{x^{2}}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7MPPA0PsU1dpF0ZPHOwf2N9SUzgALSYbl/aGC/IIM1p+rju+5Z51e/ZZSD3gRHwjBb99ixNf8xygcQ1JySdWUentN0aAmwaJ0/Zz0jfC2PcTxi2FAvU9I2PcaucfrzxFSQb2CQJd2BC0nM53oKErhOgD+MHlEtY8j9TbXGUSRhH4WRUPDnIQXYMTDiaxDMpN7" } }, { "type": "step", "result": "=\\lim_{x\\to\\:0-}\\left(e^{x}\\frac{1}{x^{2}}\\right)" }, { "type": "step", "primary": "$$\\lim_{x\\to{a}}[f\\left(x\\right)\\cdot{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\cdot\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>With the exception of indeterminate form", "result": "=\\lim_{x\\to\\:0-}\\left(e^{x}\\right)\\cdot\\:\\lim_{x\\to\\:0-}\\left(\\frac{1}{x^{2}}\\right)", "meta": { "title": { "extension": "Indeterminate Forms:<br/>$$\\frac{\\pm\\infty}{\\pm\\infty}$$<br/>$$\\frac{0}{0}$$<br/>$$\\pm\\infty\\cdot0$$<br/>$$0^0$$<br/>$$1^{\\pm\\infty}$$<br/>$$\\infty^{0}$$<br/>$$\\infty-\\infty$$" } } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:0-}\\left(e^{x}\\right)=1$$", "input": "\\lim_{x\\to\\:0-}\\left(e^{x}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=0$$", "result": "=e^{0}", "meta": { "title": { "extension": "Limit properties - if the limit of f(x), and g(x) exists, then:<br/>$$\\bullet\\quad\\lim_{x\\to\\:a}\\left(x\\right)=a$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[\\left(f\\left(x\\right)\\right)^c]=\\left(\\lim_{x\\to{a}}{f\\left(x\\right)}\\right)^c$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\pm{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\pm\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\cdot{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\cdot\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:$$where $$\\lim_{x\\to{a}}g\\left(x\\right)\\neq0$$" } } }, { "type": "step", "primary": "Simplify", "result": "=1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:0-}\\left(\\frac{1}{x^{2}}\\right)=\\infty\\:$$", "input": "\\lim_{x\\to\\:0-}\\left(\\frac{1}{x^{2}}\\right)", "steps": [ { "type": "step", "primary": "For $$x\\:$$approaching $$0\\:$$from the left$$,\\:x<0\\quad\\Rightarrow\\quad\\:x^{2}>0$$", "secondary": [ "The denominator is a positive quantity approaching 0 from the right" ], "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=1\\cdot\\:\\infty\\:" }, { "type": "step", "primary": "Apply Infinity Property: $$c\\cdot\\infty=\\infty$$", "result": "=\\infty\\:", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:0+}\\left(\\frac{e^{x}}{x^{2}}\\right)=\\infty\\:$$", "input": "\\lim_{x\\to\\:0+}\\left(\\frac{e^{x}}{x^{2}}\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{e^{x}}{x^{2}}=e^{x}\\frac{1}{x^{2}}$$", "input": "\\frac{e^{x}}{x^{2}}", "steps": [ { "type": "step", "primary": "Rewrite $$e^{x}$$ as $$e^{x}\\cdot\\:1$$", "result": "=\\frac{e^{x}\\cdot\\:1}{x^{2}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a*b}{c}=a*\\frac{b}{c}$$", "secondary": [ "$$\\frac{e^{x}\\cdot\\:1}{x^{2}}=e^{x}\\frac{1}{x^{2}}$$" ], "result": "=e^{x}\\frac{1}{x^{2}}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7MPPA0PsU1dpF0ZPHOwf2N9SUzgALSYbl/aGC/IIM1p+rju+5Z51e/ZZSD3gRHwjBb99ixNf8xygcQ1JySdWUentN0aAmwaJ0/Zz0jfC2PcTxi2FAvU9I2PcaucfrzxFSQb2CQJd2BC0nM53oKErhOgD+MHlEtY8j9TbXGUSRhH4WRUPDnIQXYMTDiaxDMpN7" } }, { "type": "step", "result": "=\\lim_{x\\to\\:0+}\\left(e^{x}\\frac{1}{x^{2}}\\right)" }, { "type": "step", "primary": "$$\\lim_{x\\to{a}}[f\\left(x\\right)\\cdot{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\cdot\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>With the exception of indeterminate form", "result": "=\\lim_{x\\to\\:0+}\\left(e^{x}\\right)\\cdot\\:\\lim_{x\\to\\:0+}\\left(\\frac{1}{x^{2}}\\right)", "meta": { "title": { "extension": "Indeterminate Forms:<br/>$$\\frac{\\pm\\infty}{\\pm\\infty}$$<br/>$$\\frac{0}{0}$$<br/>$$\\pm\\infty\\cdot0$$<br/>$$0^0$$<br/>$$1^{\\pm\\infty}$$<br/>$$\\infty^{0}$$<br/>$$\\infty-\\infty$$" } } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:0+}\\left(e^{x}\\right)=1$$", "input": "\\lim_{x\\to\\:0+}\\left(e^{x}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=0$$", "result": "=e^{0}", "meta": { "title": { "extension": "Limit properties - if the limit of f(x), and g(x) exists, then:<br/>$$\\bullet\\quad\\lim_{x\\to\\:a}\\left(x\\right)=a$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[\\left(f\\left(x\\right)\\right)^c]=\\left(\\lim_{x\\to{a}}{f\\left(x\\right)}\\right)^c$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\pm{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\pm\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\cdot{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\cdot\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:$$where $$\\lim_{x\\to{a}}g\\left(x\\right)\\neq0$$" } } }, { "type": "step", "primary": "Simplify", "result": "=1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:0+}\\left(\\frac{1}{x^{2}}\\right)=\\infty\\:$$", "input": "\\lim_{x\\to\\:0+}\\left(\\frac{1}{x^{2}}\\right)", "steps": [ { "type": "step", "primary": "For $$x\\:$$approaching $$0\\:$$from the right$$,\\:x>0\\quad\\Rightarrow\\quad\\:x^{2}>0$$", "secondary": [ "The denominator is a positive quantity approaching 0 from the right" ], "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=1\\cdot\\:\\infty\\:" }, { "type": "step", "primary": "Apply Infinity Property: $$c\\cdot\\infty=\\infty$$", "result": "=\\infty\\:", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "\\mathrm{True}" } ], "meta": { "interimType": "Vertical Asymptotes One 1Eq" } }, { "type": "step", "primary": "The vertical asymptotes are:", "result": "x=0" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Vertical Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OM+9IkDmC/hwmX5Axiwqw5dVnUL0bpHaJzHH2MFgu3oeKmDc+jjP/tiKax9A8yYYueRSpN33oxZMojoqvYhvSJAMBxyAX3aEQ7z5W7QbMWGScOIBjVD0w41lRvWu8QFmOAkNnRdvtX95Yb40vf1eN5WBZFQ8OchBdgxMOJrEMyk3s=" } }, { "type": "interim", "title": "Horizontal Asymptotes of $$\\frac{e^{x}}{x^{2}}:{\\quad}y=0$$", "input": "\\frac{e^{x}}{x^{2}}", "steps": [ { "type": "step", "primary": "Check if at $$x\\to\\pm\\infty$$ the function $$y=\\frac{e^{x}}{x^{2}}$$ behaves as a line, $$y=b$$" }, { "type": "interim", "title": "Find an asymptote for $$x\\to-\\infty\\::{\\quad}y=0$$", "steps": [ { "type": "step", "primary": "Compute $$\\lim_{x\\to-\\infty\\:}{f\\left(x\\right)}\\:$$to find b:" }, { "type": "interim", "title": "$$\\lim_{x\\to-\\infty\\:}{f\\left(x\\right)}=\\lim_{x\\to-\\infty\\:}{\\frac{e^{x}}{x^{2}}}=0$$", "input": "\\lim_{x\\to\\:-\\infty\\:}\\left(\\frac{e^{x}}{x^{2}}\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}[\\frac{f\\left(x\\right)}{g\\left(x\\right)}]=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:\\quad\\lim_{x\\to{a}}{g\\left(x\\right)}\\neq0$$<br/>With the exception of indeterminate form", "result": "=\\frac{\\lim_{x\\to\\:-\\infty\\:}\\left(e^{x}\\right)}{\\lim_{x\\to\\:-\\infty\\:}\\left(x^{2}\\right)}", "meta": { "title": { "extension": "Indeterminate Forms:<br/>$$\\frac{\\pm\\infty}{\\pm\\infty}$$<br/>$$\\frac{0}{0}$$<br/>$$\\pm\\infty\\cdot0$$<br/>$$0^0$$<br/>$$1^{\\pm\\infty}$$<br/>$$\\infty^{0}$$<br/>$$\\infty-\\infty$$" } } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:-\\infty\\:}\\left(e^{x}\\right)=0$$", "input": "\\lim_{x\\to\\:-\\infty\\:}\\left(e^{x}\\right)", "steps": [ { "type": "step", "primary": "Apply the common limit: $$\\lim_{x\\to\\:-\\infty\\:}\\left(e^{x}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:-\\infty\\:}\\left(x^{2}\\right)=\\infty\\:$$", "input": "\\lim_{x\\to\\:-\\infty\\:}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$\\lim_{x\\to\\pm\\infty}\\left(ax^{n}+\\cdots+bx+c\\right)=\\infty,\\:a>0,\\:$$n is even<br/>$$a=1,\\:n=2$$", "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+jbUbsVfpMbXIJ0UHfO5sW8yQ0stCqMVssrXgR2l7RB2D7i9sYpm+RqCkMjInr+m9btD533cgzHO58Vzwl73PAudFfSb5EGya/un1pgRn2a0wKciUtQg/2njDjXZOJLx5pYbHEl1zYMbNoc/CxPb8Tmqm40qzyGVY+WsRGEnpKJarGDlHNgwSdrYmV8S+RDA+D9a+DqAkCA8aT0cL1QXCtbEntzkBfjeX+WSvPaeRIo=" } }, { "type": "step", "result": "=\\frac{0}{\\infty\\:}" }, { "type": "step", "primary": "Apply Infinity Property: $$\\frac{0}{\\infty\\:}=0$$", "result": "=0", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "\\mathrm{No\\:horizontal\\:asymptote}" } ], "meta": { "interimType": "Horizontal Asymptotes Side 1Eq" } }, { "type": "interim", "title": "Find an asymptote for $$x\\to\\infty\\::{\\quad}$$None", "steps": [ { "type": "step", "primary": "Compute $$\\lim_{x\\to\\infty\\:}{f\\left(x\\right)}\\:$$to find b:" }, { "type": "interim", "title": "$$\\lim_{x\\to\\infty\\:}{f\\left(x\\right)}=\\lim_{x\\to\\infty\\:}{\\frac{e^{x}}{x^{2}}}=\\infty\\:$$", "input": "\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{e^{x}}{x^{2}}\\right)", "steps": [ { "type": "interim", "title": "Apply L'Hopital's Rule", "input": "\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{e^{x}}{x^{2}}\\right)", "result": "=\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{e^{x}}{2x}\\right)", "steps": [ { "type": "definition", "title": "L'Hopital Theorem:", "text": "For $$\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right),\\:\\mathrm{if}\\:\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{0}{0}\\quad\\mathrm{or}\\quad\\lim_{x\\to\\:a}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{\\pm\\infty}{\\pm\\infty},\\:\\mathrm{then}$$<br/>$$\\quad\\quad\\quad\\bold{\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\lim_{x\\to{a}}\\left(\\frac{f^{'}\\left(x\\right)}{g^{'}\\left(x\\right)}\\right)}$$" }, { "type": "interim", "title": "Test L'Hopital condition: $$\\frac{\\infty\\:}{\\infty\\:}$$", "result": "=\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{\\left(e^{x}\\right)^{^{\\prime}}}{\\left(x^{2}\\right)^{^{\\prime}}}\\right)", "steps": [ { "type": "interim", "title": "$$\\lim_{x\\to\\:\\infty\\:}\\left(e^{x}\\right)=\\infty\\:$$", "input": "\\lim_{x\\to\\:\\infty\\:}\\left(e^{x}\\right)", "steps": [ { "type": "step", "primary": "Apply the common limit: $$\\lim_{x\\to\\:\\infty\\:}\\left(e^{x}\\right)=\\infty\\:$$", "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:\\infty\\:}\\left(x^{2}\\right)=\\infty\\:$$", "input": "\\lim_{x\\to\\:\\infty\\:}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$\\lim_{x\\to\\pm\\infty}\\left(ax^{n}+\\cdots+bx+c\\right)=\\infty,\\:a>0,\\:$$n is even<br/>$$a=1,\\:n=2$$", "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+jbUbsVfpMbXIJ0UHfO5saL0uDZ6PHhP1GqJznWyHwkaMS1C7LHR3Ox3wzQnf9x+cAgzx9zrYTFpwIg0trnpZTVTxY0hilST8p7oi8GAvx7QF2UzOp5Bb4XpeTJ7DlfnwhWwQYyckKycWgIwD4f3sBi2Sg2N1jZXcumfy0+UpgCi9Lg2ejx4T9Rqic51sh8JeOKMC/3/n4eO0zNsI03rcdnUMsHFIy5Nsmxty17Bu2Q=" } }, { "type": "step", "primary": "Meets L'hopital condition of: $$\\frac{\\infty\\:}{\\infty\\:}$$", "result": "=\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{\\left(e^{x}\\right)^{^{^{\\prime}}}}{\\left(x^{2}\\right)^{^{^{\\prime}}}}\\right)" } ], "meta": { "interimType": "Lhopital Condition 1Eq" } }, { "type": "interim", "title": "$$\\left(e^{x}\\right)'=e^{x}$$", "input": "\\frac{d}{dx}\\left(e^{x}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(e^{x}\\right)=e^{x}$$", "result": "=e^{x}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjqxKAa6SkUhZrTPjmns35ik3hxk9aCfAWodBRxXgUexthpiW0WhiZGad41dobHknD/L0MoYg+CUn6oyL3EO7Yp1B2hMQUtdwgYJZ1r+cLQNJLd1ohke2Wgml78++2zI0g==" } }, { "type": "interim", "title": "$$\\left(x^{2}\\right)'=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkmb3s5xAUYje7fZWSRkdb2k3hxk9aCfAWodBRxXgUexcQsmN/cITrVSOMImEqe3fudHvcmjs8iJXuaQ+nxJX156Xz0OParmIowNHqIzDBN8" } }, { "type": "step", "result": "=\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{e^{x}}{2x}\\right)" } ], "meta": { "interimType": "Lhopital Theorem 0Eq", "practiceLink": "/practice/limits-practice#area=main&subtopic=L'Hopital%20Rule", "practiceTopic": "L'hopital Rule", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+jbUbsVfpMbXIJ0UHfO5saL0uDZ6PHhP1GqJznWyHwmWn2tyts4ihSIs6Ah0p2yXtcngnTW1AWFeb02vni0XLbmIKnaLGSq3SuzYRYyGsT2mmyLIEdPsejKo8OmKWvZOgSEHRda+G5BHM5FRE2/NG+7LUvr52HEm2rBNGzBdDyCk+q2xbw+ySI/eV65OSy9291sIFIsmZBinqaLjpKnVeU3kCh3oevUunZ7/b0qFKBSyIAv17r6SEQ+f7Q/d2FLEdYUDEApzDR4jzQUGsjov1w==" } }, { "type": "interim", "title": "Apply L'Hopital's Rule", "input": "\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{e^{x}}{2x}\\right)", "result": "=\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{e^{x}}{2}\\right)", "steps": [ { "type": "definition", "title": "L'Hopital Theorem:", "text": "For $$\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right),\\:\\mathrm{if}\\:\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{0}{0}\\quad\\mathrm{or}\\quad\\lim_{x\\to\\:a}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{\\pm\\infty}{\\pm\\infty},\\:\\mathrm{then}$$<br/>$$\\quad\\quad\\quad\\bold{\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\lim_{x\\to{a}}\\left(\\frac{f^{'}\\left(x\\right)}{g^{'}\\left(x\\right)}\\right)}$$" }, { "type": "interim", "title": "Test L'Hopital condition: $$\\frac{\\infty\\:}{\\infty\\:}$$", "result": "=\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{\\left(e^{x}\\right)^{^{\\prime}}}{\\left(2x\\right)^{^{\\prime}}}\\right)", "steps": [ { "type": "interim", "title": "$$\\lim_{x\\to\\:\\infty\\:}\\left(e^{x}\\right)=\\infty\\:$$", "input": "\\lim_{x\\to\\:\\infty\\:}\\left(e^{x}\\right)", "steps": [ { "type": "step", "primary": "Apply the common limit: $$\\lim_{x\\to\\:\\infty\\:}\\left(e^{x}\\right)=\\infty\\:$$", "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:\\infty\\:}\\left(2x\\right)=\\infty\\:$$", "input": "\\lim_{x\\to\\:\\infty\\:}\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$\\lim_{x\\to\\infty}\\left(ax^{n}+\\cdots+bx+c\\right)=\\infty,\\:a>0,\\:$$n is odd<br/>$$a=2,\\:n=1$$", "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+jbUbsVfpMbXIJ0UHfO5saL0uDZ6PHhP1GqJznWyHwlV+qg7Q+0fNarYODiCp208gWZR3zCAilVrsafPmsT5abA9Nb3zLvO1gWJufFEd2tIwFfVA2P3O5Qj7EDGNsqJdP3yb3suSQQSaGuJSfY5SMAaS5CGs21zxA42OGelXxJHeyE0iYvjxncDM4tBgVfPAyD8MhUKmOPCU9J4rK7X6Ug==" } }, { "type": "step", "primary": "Meets L'hopital condition of: $$\\frac{\\infty\\:}{\\infty\\:}$$", "result": "=\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{\\left(e^{x}\\right)^{^{^{\\prime}}}}{\\left(2x\\right)^{^{^{\\prime}}}}\\right)" } ], "meta": { "interimType": "Lhopital Condition 1Eq" } }, { "type": "interim", "title": "$$\\left(e^{x}\\right)'=e^{x}$$", "input": "\\frac{d}{dx}\\left(e^{x}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(e^{x}\\right)=e^{x}$$", "result": "=e^{x}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjqxKAa6SkUhZrTPjmns35ik3hxk9aCfAWodBRxXgUexthpiW0WhiZGad41dobHknD/L0MoYg+CUn6oyL3EO7Yp1B2hMQUtdwgYJZ1r+cLQNJLd1ohke2Wgml78++2zI0g==" } }, { "type": "interim", "title": "$$\\left(2x\\right)'=2$$", "input": "\\frac{d}{dx}\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYg2sQzwGEAAPyDk8n13Ps8XZGku9zFkxwe1dTH8vycb94wHsFp27x8BxzSfXYcuPliXKvlioccNR9e+UjAgV/3qwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{e^{x}}{2}\\right)" } ], "meta": { "interimType": "Lhopital Theorem 0Eq", "practiceLink": "/practice/limits-practice#area=main&subtopic=L'Hopital%20Rule", "practiceTopic": "L'hopital Rule", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+jbUbsVfpMbXIJ0UHfO5saL0uDZ6PHhP1GqJznWyHwmWn2tyts4ihSIs6Ah0p2yXChKL+0uARTKwznN1sBA+oljlDP5aX4SprmJ1raef0WDidrg8DC82xolA3cBpRMkxM13xa8ZsWq5c7JLPonGhI7OhsTL2emzAKqGHxCz6tu2b4cvchfV+5bNofnpxdk5a35owguej18ZHfjpsA1+YiAS4M5VpC8qh+oehjmM1qmwtqhtTKB1YevrvLbYub3J5BnQ/fS3TdY5cJWNofcUKJg==" } }, { "type": "step", "primary": "$$\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$", "result": "=\\frac{1}{2}\\cdot\\:\\lim_{x\\to\\:\\infty\\:}\\left(e^{x}\\right)" }, { "type": "step", "primary": "Apply the common limit: $$\\lim_{x\\to\\:\\infty\\:}\\left(e^{x}\\right)=\\infty\\:$$", "result": "=\\frac{1}{2}\\cdot\\:\\infty\\:" }, { "type": "step", "primary": "Apply Infinity Property: $$c\\cdot\\infty=\\infty$$", "result": "=\\infty\\:", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "primary": "The result is not a finite constant, therefore" }, { "type": "step", "result": "\\mathrm{No\\:Horizontal\\:Asymptote}" } ], "meta": { "interimType": "Horizontal Asymptotes Side 1Eq" } }, { "type": "step", "primary": "The horizontal asymptote is:", "result": "y=0" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Horizontal Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMW0QiF7WQsnIxF55f0PihooH8V6rf2M5+iqpeDLe57MujlcvuLs2HsfPHylPaOYiCZEt3ZXAiqUE0HIXrrrezJO7tpg6+wppGIR16yRwSoZwhY1KWFY4dVpjknbwHn5uVvdhfqp0iMdPvHHBbjHl0zFypnVfW1elSOrolhvXW8DI=" } }, { "type": "interim", "title": "Slant Asymptotes of $$\\frac{e^{x}}{x^{2}}:{\\quad}$$None", "input": "\\frac{e^{x}}{x^{2}}", "steps": [ { "type": "step", "primary": "Check if at $$x\\to\\pm\\infty$$ the function $$y=\\frac{e^{x}}{x^{2}}$$ behaves as a line, $$y=mx+b\\:$$ where $$m\\neq0$$" }, { "type": "interim", "title": "Find an asymptote for $$x\\to-\\infty\\::{\\quad}$$None", "steps": [ { "type": "step", "primary": "Compute $$\\lim_{x\\to-\\infty\\:}{\\frac{f\\left(x\\right)}{x}}\\:$$to find m:" }, { "type": "interim", "title": "$$\\lim_{x\\to-\\infty\\:}{\\frac{f\\left(x\\right)}{x}}=\\lim_{x\\to-\\infty\\:}{\\frac{\\frac{e^{x}}{x^{2}}}{x}}=0$$", "input": "\\lim_{x\\to\\:-\\infty\\:}\\left(\\frac{\\frac{e^{x}}{x^{2}}}{x}\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}[\\frac{f\\left(x\\right)}{g\\left(x\\right)}]=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:\\quad\\lim_{x\\to{a}}{g\\left(x\\right)}\\neq0$$<br/>With the exception of indeterminate form", "result": "=\\frac{\\lim_{x\\to\\:-\\infty\\:}\\left(\\frac{e^{x}}{x^{2}}\\right)}{\\lim_{x\\to\\:-\\infty\\:}\\left(x\\right)}", "meta": { "title": { "extension": "Indeterminate Forms:<br/>$$\\frac{\\pm\\infty}{\\pm\\infty}$$<br/>$$\\frac{0}{0}$$<br/>$$\\pm\\infty\\cdot0$$<br/>$$0^0$$<br/>$$1^{\\pm\\infty}$$<br/>$$\\infty^{0}$$<br/>$$\\infty-\\infty$$" } } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:-\\infty\\:}\\left(\\frac{e^{x}}{x^{2}}\\right)=0$$", "input": "\\lim_{x\\to\\:-\\infty\\:}\\left(\\frac{e^{x}}{x^{2}}\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}[\\frac{f\\left(x\\right)}{g\\left(x\\right)}]=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:\\quad\\lim_{x\\to{a}}{g\\left(x\\right)}\\neq0$$<br/>With the exception of indeterminate form", "result": "=\\frac{\\lim_{x\\to\\:-\\infty\\:}\\left(e^{x}\\right)}{\\lim_{x\\to\\:-\\infty\\:}\\left(x^{2}\\right)}", "meta": { "title": { "extension": "Indeterminate Forms:<br/>$$\\frac{\\pm\\infty}{\\pm\\infty}$$<br/>$$\\frac{0}{0}$$<br/>$$\\pm\\infty\\cdot0$$<br/>$$0^0$$<br/>$$1^{\\pm\\infty}$$<br/>$$\\infty^{0}$$<br/>$$\\infty-\\infty$$" } } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:-\\infty\\:}\\left(e^{x}\\right)=0$$", "input": "\\lim_{x\\to\\:-\\infty\\:}\\left(e^{x}\\right)", "steps": [ { "type": "step", "primary": "Apply the common limit: $$\\lim_{x\\to\\:-\\infty\\:}\\left(e^{x}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:-\\infty\\:}\\left(x^{2}\\right)=\\infty\\:$$", "input": "\\lim_{x\\to\\:-\\infty\\:}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$\\lim_{x\\to\\pm\\infty}\\left(ax^{n}+\\cdots+bx+c\\right)=\\infty,\\:a>0,\\:$$n is even<br/>$$a=1,\\:n=2$$", "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+jbUbsVfpMbXIJ0UHfO5sW8yQ0stCqMVssrXgR2l7RB2D7i9sYpm+RqCkMjInr+m9btD533cgzHO58Vzwl73PAudFfSb5EGya/un1pgRn2a0wKciUtQg/2njDjXZOJLx5pYbHEl1zYMbNoc/CxPb8Tmqm40qzyGVY+WsRGEnpKJarGDlHNgwSdrYmV8S+RDA+D9a+DqAkCA8aT0cL1QXCtbEntzkBfjeX+WSvPaeRIo=" } }, { "type": "step", "result": "=\\frac{0}{\\infty\\:}" }, { "type": "step", "primary": "Apply Infinity Property: $$\\frac{0}{\\infty\\:}=0$$", "result": "=0", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:-\\infty\\:}\\left(x\\right)=-\\infty\\:$$", "input": "\\lim_{x\\to\\:-\\infty\\:}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply the common limit: $$\\lim_{x\\to\\:-\\infty\\:}\\left(x\\right)=-\\infty\\:$$", "result": "=-\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=\\frac{0}{-\\infty\\:}" }, { "type": "step", "primary": "Apply Infinity Property: $$\\frac{0}{-\\infty\\:}=0$$", "result": "=0", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "primary": "The slope is zero, therefore" }, { "type": "step", "result": "\\mathrm{No\\:slant\\:asymptote}" } ], "meta": { "interimType": "Horizontal Asymptotes Side 1Eq" } }, { "type": "interim", "title": "Find an asymptote for $$x\\to\\infty\\::{\\quad}$$None", "steps": [ { "type": "step", "primary": "Compute $$\\lim_{x\\to\\infty\\:}{\\frac{f\\left(x\\right)}{x}}\\:$$to find m:" }, { "type": "interim", "title": "$$\\lim_{x\\to\\infty\\:}{\\frac{f\\left(x\\right)}{x}}=\\lim_{x\\to\\infty\\:}{\\frac{\\frac{e^{x}}{x^{2}}}{x}}=\\infty\\:$$", "input": "\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{\\frac{e^{x}}{x^{2}}}{x}\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{\\frac{e^{x}}{x^{2}}}{x}:{\\quad}\\frac{e^{x}}{x^{3}}$$", "input": "\\frac{\\frac{e^{x}}{x^{2}}}{x}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{e^{x}}{x^{2}x}" }, { "type": "interim", "title": "$$x^{2}x=x^{3}$$", "input": "x^{2}x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$x^{2}x=\\:x^{2+1}$$" ], "result": "=x^{2+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=x^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7LWrIAcT/Pu2l3a2UlrsCC3WD310L1+P2yDQQfMEhENGDPbrpZaMLRegCZc+JnvJI5kmDAAHjIPJcICsCIhoRbVuSVZd9z4+kRKtqsjU2P18=" } }, { "type": "step", "result": "=\\frac{e^{x}}{x^{3}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{e^{x}}{x^{3}}\\right)" }, { "type": "interim", "title": "Apply L'Hopital's Rule", "input": "\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{e^{x}}{x^{3}}\\right)", "result": "=\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{e^{x}}{6}\\right)", "steps": [ { "type": "definition", "title": "L'Hopital Theorem:", "text": "For $$\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right),\\:\\mathrm{if}\\:\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{0}{0}\\quad\\mathrm{or}\\quad\\lim_{x\\to\\:a}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{\\pm\\infty}{\\pm\\infty},\\:\\mathrm{then}$$<br/>$$\\quad\\quad\\quad\\bold{\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\lim_{x\\to{a}}\\left(\\frac{f^{'}\\left(x\\right)}{g^{'}\\left(x\\right)}\\right)}$$" }, { "type": "step", "primary": "Apply L'Hopital's Rule 3 times" }, { "type": "interim", "title": "Test L'Hopital condition: $$\\frac{\\infty\\:}{\\infty\\:}$$", "result": "=\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{\\frac{d^{3}}{d^{3}x}\\left(e^{x}\\right)}{\\frac{d^{3}}{d^{3}x}\\left(x^{3}\\right)}\\right)", "steps": [ { "type": "step", "primary": "For each $$n\\in\\{1..3\\}\\:$$", "secondary": [ "$$\\lim{x\\to\\infty}\\left(\\frac{d^n}{d^{n}x}\\left(e^{x}\\right)\\right)=\\infty$$", "$$\\lim{x\\to\\infty}\\left(\\frac{d^n}{d^{n}x}\\left(x^{n}\\right)\\right)=\\infty$$" ] }, { "type": "step", "primary": "Meets L'hopital condition of: $$\\frac{\\infty\\:}{\\infty\\:}$$", "result": "=\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{\\frac{d^{3}}{d^{3}x}\\left(e^{x}\\right)}{\\frac{d^{3}}{d^{3}x}\\left(x^{3}\\right)}\\right)" } ], "meta": { "interimType": "Lhopital Condition 1Eq" } }, { "type": "interim", "title": "$$\\frac{d^{3}}{dx^{3}}\\left(e^{x}\\right)=e^{x}$$", "steps": [ { "type": "step", "primary": "Apply the Nth derivative rule: $$\\frac{d^{n}}{d^{n}x}\\left(e^{x}\\right)=e^{x}$$", "result": "=e^{x}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\frac{d^{3}}{dx^{3}}\\left(x^{3}\\right)=\\frac{3!}{\\left(3-3\\right)!}x^{3-3}$$", "steps": [ { "type": "step", "primary": "Apply the Nth derivative rule: $$\\frac{d^{n}}{d^{n}x}\\left(x^{a}\\right)=\\frac{a!}{\\left(a-n\\right)!}x^{a-n}$$", "result": "=\\frac{3!}{\\left(3-3\\right)!}x^{3-3}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\lim_{x\\to\\:\\infty\\:}\\left(\\frac{e^{x}}{\\frac{3!}{\\left(3-3\\right)!}x^{3-3}}\\right)" }, { "type": "interim", "title": "Simplify $$\\frac{e^{x}}{\\frac{3!}{\\left(3-3\\right)!}x^{3-3}}:{\\quad}\\frac{e^{x}}{6}$$", "input": "\\frac{e^{x}}{\\frac{3!}{\\left(3-3\\right)!}x^{3-3}}", "steps": [ { "type": "interim", "title": "$$\\frac{3!}{\\left(3-3\\right)!}x^{3-3}=3!$$", "input": "\\frac{3!}{\\left(3-3\\right)!}x^{3-3}", "steps": [ { "type": "interim", "title": "$$\\frac{3!}{\\left(3-3\\right)!}=3!$$", "input": "\\frac{3!}{\\left(3-3\\right)!}", "steps": [ { "type": "interim", "title": "$$\\left(3-3\\right)!=1$$", "input": "\\left(3-3\\right)!", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$3-3=0$$", "result": "=0!" }, { "type": "step", "primary": "Apply factorial rule: $$0!=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lKDPNxbaMfdPJAyvfBnpM3WD310L1+P2yDQQfMEhENH7xkO7yIXgxJ7C3jKazWUh8lXmv4lP6P7Jvj4aXeWD0iS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=\\frac{3!}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=3!" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+sA4swU7K28S9Y/D1El+Lv9tm4+/YhpkAalqPeB/Tm/MwViaLUXkeD+JukROhWdjb++XVBrsF+wY0lqbOL/SlCOdSvUsGXX86n27SUtSCSxtwy3Fx0TKYG+mcA3Gp+HG" } }, { "type": "step", "result": "=3!x^{3-3}" }, { "type": "interim", "title": "$$x^{3-3}=1$$", "input": "x^{3-3}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$3-3=0$$", "result": "=x^{0}" }, { "type": "step", "primary": "Apply rule $$a^{0}=1,\\:a\\ne\\:0$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VKM753yISqbLTrUWJ3qG2gOfOVs9mPIqDLV5QIWwt3mXUrWvRppVP8TTNlqQgP2Mhmv4pmQPY5i5qFEL0hB6KrCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=1\\cdot\\:3!" }, { "type": "step", "primary": "Multiply: $$3!\\cdot\\:1=3!$$", "result": "=3!" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+sA4swU7K28S9Y/D1El+Lqzj+HDrtE/SV+IhKUyTuUndd47a0hQ8flDbGsI5To1dJFofktfvXIQ8Ea93Pv5YWs1bIZxfodm3UsZcfZAZr4sg8M4bdWqRcs8YaH8Ts4Jxd3S/1ejbi5GXoeGnJhX8Hg==" } }, { "type": "step", "result": "=\\frac{e^{x}}{3!}" }, { "type": "interim", "title": "$$3!=6$$", "input": "3!", "steps": [ { "type": "step", "primary": "Apply factorial rule: $$n!=1\\cdot2\\cdot3\\cdot\\ldots\\cdot\\:n$$", "secondary": [ "$$3!=1\\cdot\\:2\\cdot\\:3$$" ], "result": "=1\\cdot\\:2\\cdot\\:3" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2\\cdot\\:3=6$$", "result": "=6" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7RciCdt6M4n7wesJ3iBJPn8zBWJotReR4P4m6RE6FZ2No8psdWgO5IuIgJajcfvo9neavaRADROt2EOT6omsR8g==" } }, { "type": "step", "result": "=\\frac{e^{x}}{6}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{e^{x}}{6}" } ], "meta": { "interimType": "Lhopital Theorem 0Eq", "practiceLink": "/practice/limits-practice#area=main&subtopic=L'Hopital%20Rule", "practiceTopic": "L'hopital Rule", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+jbUbsVfpMbXIJ0UHfO5saL0uDZ6PHhP1GqJznWyHwmWn2tyts4ihSIs6Ah0p2yXRx9E8FbWChpsE7rQANkqJ7mIKnaLGSq3SuzYRYyGsT2mmyLIEdPsejKo8OmKWvZOgSEHRda+G5BHM5FRE2/NG6sB5dNg6RVDtv4jBiEqgidFKk3fejFkyiOiq9iG9IkA2tYMfhBNM0rRjNP9zHHB8rcFLDuhsiPQNMF+op0gFSk=" } }, { "type": "step", "primary": "$$\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$", "result": "=\\frac{1}{6}\\cdot\\:\\lim_{x\\to\\:\\infty\\:}\\left(e^{x}\\right)" }, { "type": "step", "primary": "Apply the common limit: $$\\lim_{x\\to\\:\\infty\\:}\\left(e^{x}\\right)=\\infty\\:$$", "result": "=\\frac{1}{6}\\cdot\\:\\infty\\:" }, { "type": "step", "primary": "Apply Infinity Property: $$c\\cdot\\infty=\\infty$$", "result": "=\\infty\\:", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "primary": "The slope is not a finite constant, therefore" }, { "type": "step", "result": "\\mathrm{No\\:slant\\:asymptote}" } ], "meta": { "interimType": "Horizontal Asymptotes Side 1Eq" } }, { "type": "step", "result": "\\mathrm{No\\:slant\\:asymptote}" } ], "meta": { "interimType": "Slant Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMLXiD7VTAsp80tg/tmv96KCiDolSLDvdTzf0ZC7B11gedRM5YY7QLl/mL7uuBQjAL8LfSxJ+0AgVLpCSnLX0iSgI5i9maKPYxQTldBPZ3WKHkjoFKINMdqnHv78/vwbTsZIX+xVTqOg5rVnIKmOyioQ==" } }, { "type": "step", "result": "\\mathrm{Vertical}:\\:x=0,\\:\\mathrm{Horizontal}:\\:y=0" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Function Asymptotes Top 2Eq" } }, { "type": "interim", "title": "Extreme Points of $$\\frac{e^{x}}{x^{2}}:{\\quad}$$Minimum$$\\left(2,\\:\\frac{e^{2}}{4}\\right)$$", "steps": [ { "type": "definition", "title": "First Derivative Test definition", "text": "Suppose that $$x=c$$ is a critical point of $$f\\left(x\\right)$$ then, <br/>If $$f\\:{^{\\prime}}\\left(x\\right)>0$$ to the left of $$x=c$$ and $$f\\:{^{\\prime}}\\left(x\\right)<0$$ to the right of $$x=c$$ then $$x=c$$ is a local maximum.<br/>If $$f\\:{^{\\prime}}\\left(x\\right)<0$$ to the left of $$x=c$$ and $$f\\:{^{\\prime}}\\left(x\\right)>\\:0$$ to the right of $$x=c$$ then $$x=c$$ is a local minimum.<br/>If $$f\\:{^{\\prime}}\\left(x\\right)$$ is the same sign on both sides of $$x=c$$ then $$x=c$$ is neither a local maximum nor a local minimum." }, { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(x\\right)=\\frac{e^{x}\\left(x-2\\right)}{x^{3}}$$", "input": "\\frac{d}{dx}\\left(\\frac{e^{x}}{x^{2}}\\right)", "steps": [ { "type": "step", "primary": "Apply the Quotient Rule: $$\\left(\\frac{f}{g}\\right)'=\\frac{f'{\\cdot}g-g'{\\cdot}f}{g^{2}}$$", "result": "=\\frac{\\frac{d}{dx}\\left(e^{x}\\right)x^{2}-\\frac{d}{dx}\\left(x^{2}\\right)e^{x}}{\\left(x^{2}\\right)^{2}}" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(e^{x}\\right)=e^{x}$$", "input": "\\frac{d}{dx}\\left(e^{x}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(e^{x}\\right)=e^{x}$$", "result": "=e^{x}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjqxKAa6SkUhZrTPjmns35ik3hxk9aCfAWodBRxXgUexthpiW0WhiZGad41dobHknD/L0MoYg+CUn6oyL3EO7YobH0DM3/kWbWNIKRYne2WotxxYM+bRuETn+hrcvGmLsg==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkmb3s5xAUYje7fZWSRkdb2k3hxk9aCfAWodBRxXgUexcQsmN/cITrVSOMImEqe3fkeCBKuYKgaNJ253gLI69U7cjrVUqImvoUuRtb+2ccCzWsr9JoDNJaP7hueshcYJ6w==" } }, { "type": "step", "result": "=\\frac{e^{x}x^{2}-2xe^{x}}{\\left(x^{2}\\right)^{2}}" }, { "type": "interim", "title": "Simplify $$\\frac{e^{x}x^{2}-2xe^{x}}{\\left(x^{2}\\right)^{2}}:{\\quad}\\frac{e^{x}\\left(x-2\\right)}{x^{3}}$$", "input": "\\frac{e^{x}x^{2}-2xe^{x}}{\\left(x^{2}\\right)^{2}}", "result": "=\\frac{e^{x}\\left(x-2\\right)}{x^{3}}", "steps": [ { "type": "interim", "title": "Factor $$e^{x}x^{2}-2xe^{x}:{\\quad}xe^{x}\\left(x-2\\right)$$", "input": "e^{x}x^{2}-2xe^{x}", "result": "=\\frac{xe^{x}\\left(x-2\\right)}{\\left(x^{2}\\right)^{2}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=e^{x}xx-2xe^{x}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$xe^{x}$$", "result": "=xe^{x}\\left(x-2\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Simplify $$\\left(x^{2}\\right)^{2}:{\\quad}x^{4}$$", "input": "\\left(x^{2}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=x^{2\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=x^{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jd2F9SNQ4l8Y5LjObCinw96GQqufR6tr2vPxOUv7H+9UXCH/RczT7MepT8JUGxNwZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz13rvDT7XPh5g1GffZc9C9/" } }, { "type": "step", "result": "=\\frac{e^{x}x\\left(x-2\\right)}{x^{4}}" }, { "type": "step", "primary": "Cancel the common factor: $$x$$", "result": "=\\frac{e^{x}\\left(x-2\\right)}{x^{3}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GCORLVVHoNUKKIKCG8MW2BGOPVVNpdDH/kxsTI2YUVZ1q76C8OazfU2FcovI65TQA585Wz2Y8ioMtXlAhbC3eUntoSNn7Xag8VSkCd+TyFh3A+MYmDLVMW8ajGkszMT/ZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz15vgWPvL2Q7OhNZveWk3xcEY49VU2l0Mf+TGxMjZhRVkddNPdHlszcwQPnreyfM5M=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Find intervals:$${\\quad}$$Increasing$$:-\\infty\\:<x<0,\\:$$Decreasing$$:0<x<2,\\:$$Increasing$$:2<x<\\infty\\:$$", "input": "f\\:{^{\\prime}}\\left(x\\right)=\\frac{e^{x}\\left(x-2\\right)}{x^{3}}", "steps": [ { "type": "interim", "title": "Find the critical points:$${\\quad}x=0,\\:x=2$$", "steps": [ { "type": "definition", "title": "Critical point definition", "text": "Critical points are points where the function is defined and its derivative is zero or undefined" }, { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(x\\right)=0:{\\quad}x=2$$", "input": "\\frac{e^{x}\\left(x-2\\right)}{x^{3}}=0", "steps": [ { "type": "step", "primary": "$$\\frac{f\\left(x\\right)}{g\\left(x\\right)}=0{\\quad\\Rightarrow\\quad}f\\left(x\\right)=0$$", "result": "e^{x}\\left(x-2\\right)=0" }, { "type": "step", "primary": "Using the Zero Factor Principle:$$\\quad$$ If $$ab=0\\:$$then $$a=0\\:$$or $$b=0$$", "result": "e^{x}=0\\lor\\:x-2=0" }, { "type": "interim", "title": "Solve $$e^{x}=0:{\\quad}$$No Solution for $$x\\in\\mathbb{R}$$", "input": "e^{x}=0", "steps": [ { "type": "step", "primary": "$$a^{f\\left(x\\right)}$$ cannot be zero or negative for $$x\\in\\mathbb{R}$$", "result": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$x-2=0:{\\quad}x=2$$", "input": "x-2=0", "steps": [ { "type": "interim", "title": "Move $$2\\:$$to the right side", "input": "x-2=0", "result": "x=2", "steps": [ { "type": "step", "primary": "Add $$2$$ to both sides", "result": "x-2+2=0+2" }, { "type": "step", "primary": "Simplify", "result": "x=2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Verify Solutions:$${\\quad}x=2\\:$$True", "steps": [ { "type": "step", "primary": "Check the solutions by plugging them into $$\\frac{e^{x}\\left(x-2\\right)}{x^{3}}=0$$<br/>Remove the ones that don't agree with the equation." }, { "type": "interim", "title": "Plug in $$x=2:{\\quad}$$True", "input": "\\frac{e^{2}\\left(2-2\\right)}{2^{3}}=0", "steps": [ { "type": "interim", "title": "$$\\frac{e^{2}\\left(2-2\\right)}{2^{3}}=0$$", "input": "\\frac{e^{2}\\left(2-2\\right)}{2^{3}}", "steps": [ { "type": "interim", "title": "$$e^{2}\\left(2-2\\right)=0$$", "input": "e^{2}\\left(2-2\\right)", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$2-2=0$$", "result": "=0\\cdot\\:e^{2}" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7PpgbiFiym/DqgoHOnylETiAn9lkDfZkicUGkO3EF+Ip/suoH0JzK9gJYOwax5fT27/PB1ep1urYBGc+jjZ0+fg1F5/24c34qDescA26tcc0=" } }, { "type": "step", "result": "=\\frac{0}{2^{3}}" }, { "type": "step", "primary": "Apply rule $$\\frac{0}{a}=0,\\:a\\ne\\:0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WFXrvSSXKcE1RreO2RcH3SP8IK1kjM00DV1G7E+y5QPdd47a0hQ8flDbGsI5To1dTbAOxT8wOTlsw5gGf+Hdr1NbbqpyK7JQEZdATEJR51iCVEYzSCtRx3J1efhO1qcD3IXjn3aOz/DkLLPMfmW3tg==" } }, { "type": "step", "primary": "$$0=0$$" }, { "type": "step", "result": "\\mathrm{True}" } ], "meta": { "interimType": "Generic Plug 1Eq" } } ], "meta": { "interimType": "Check Solutions Plug Preface 1Eq" } }, { "type": "step", "primary": "The solution is", "result": "x=2" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "interim", "title": "Find undefined (singularity) points of $$f\\:{^{\\prime}}\\left(x\\right):{\\quad}x=0$$", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{e^{x}\\left(x-2\\right)}{x^{3}}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$x^{3}=0:{\\quad}x=0$$", "input": "x^{3}=0", "steps": [ { "type": "step", "primary": "Apply rule $$x^n=0\\quad\\Rightarrow\\quad\\:x=0$$" }, { "type": "step", "result": "x=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The following points are undefined", "result": "x=0" } ], "meta": { "interimType": "Undefined Points 0Eq" } }, { "type": "step", "result": "x=0,\\:x=2" } ], "meta": { "interimType": "Explore Function Slope Zero Title 0Eq" } }, { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(x\\right)>0:{\\quad}x<0\\lor\\:x>2$$", "input": "\\frac{e^{x}\\left(x-2\\right)}{x^{3}}>0", "steps": [ { "type": "interim", "title": "Identify the intervals", "result": "x<0\\lor\\:x>2", "steps": [ { "type": "interim", "title": "Find the signs of the factors of $$\\frac{e^{x}\\left(x-2\\right)}{x^{3}}$$", "input": "\\frac{e^{x}\\left(x-2\\right)}{x^{3}}", "steps": [ { "type": "interim", "title": "Find the signs of $$e^{x}:{\\quad}$$Positive for all real values", "input": "e^{x}>0", "steps": [ { "type": "step", "primary": "If $$a>0,\\:a^{f\\left(x\\right)}\\:$$is greater than 0", "secondary": [ "$$a=e$$" ], "result": "\\mathrm{True\\:for\\:all}\\:x\\in\\mathbb{R}" } ], "meta": { "interimType": "Find Sign 1Eq" } }, { "type": "interim", "title": "Find the signs of $$x-2$$", "steps": [ { "type": "interim", "title": "$$x-2=0:{\\quad}x=2$$", "input": "x-2=0", "steps": [ { "type": "interim", "title": "Move $$2\\:$$to the right side", "input": "x-2=0", "result": "x=2", "steps": [ { "type": "step", "primary": "Add $$2$$ to both sides", "result": "x-2+2=0+2" }, { "type": "step", "primary": "Simplify", "result": "x=2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "interim", "title": "$$x-2<0:{\\quad}x<2$$", "input": "x-2<0", "steps": [ { "type": "interim", "title": "Move $$2\\:$$to the right side", "input": "x-2<0", "result": "x<2", "steps": [ { "type": "step", "primary": "Add $$2$$ to both sides", "result": "x-2+2<0+2" }, { "type": "step", "primary": "Simplify", "result": "x<2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$x-2>0:{\\quad}x>2$$", "input": "x-2>0", "steps": [ { "type": "interim", "title": "Move $$2\\:$$to the right side", "input": "x-2>0", "result": "x>2", "steps": [ { "type": "step", "primary": "Add $$2$$ to both sides", "result": "x-2+2>0+2" }, { "type": "step", "primary": "Simplify", "result": "x>2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } } ], "meta": { "interimType": "Find Sign 1Eq" } }, { "type": "interim", "title": "Find the signs of $$x^{3}$$", "steps": [ { "type": "interim", "title": "$$x^{3}=0:{\\quad}x=0$$", "input": "x^{3}=0", "steps": [ { "type": "step", "primary": "Apply rule $$x^n=0\\quad\\Rightarrow\\quad\\:x=0$$" }, { "type": "step", "result": "x=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "interim", "title": "$$x^{3}<0:{\\quad}x<0$$", "input": "x^{3}<0", "steps": [ { "type": "step", "primary": "For $$u^n\\:<\\:0$$, if $$n\\:$$is odd then $$u\\:<\\:0$$" }, { "type": "step", "result": "x<0" } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$x^{3}>0:{\\quad}x>0$$", "input": "x^{3}>0", "steps": [ { "type": "step", "primary": "For $$u^n\\:>\\:0$$, if $$n\\:$$is odd then $$u\\:>\\:0$$" }, { "type": "step", "result": "x>0" } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } } ], "meta": { "interimType": "Find Sign 1Eq" } } ], "meta": { "interimType": "Find Signs Top 1Eq" } }, { "type": "step", "primary": "Identify the intervals:", "result": "x<0,\\:x=0,\\:0<x<2,\\:x=2,\\:x>2" }, { "type": "step", "primary": "Find singularity points" }, { "type": "interim", "title": "Find the zeros of the denominator $$x^{3}:{\\quad}x=0$$", "input": "x^{3}=0", "steps": [ { "type": "step", "primary": "Apply rule $$x^n=0\\quad\\Rightarrow\\quad\\:x=0$$" }, { "type": "step", "result": "x=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Find Denom Zeroes Title 1Eq" } }, { "type": "step", "primary": "Combine the intervals with the singularity points", "result": "x<0,\\:0<x<2,\\:x=2,\\:x>2" }, { "type": "step", "primary": "Summarize in a table:", "secondary": [ "$$\\begin{array}{|c|c|c|c|c|}\\hline &x<0&0<x<2&x=2&x>2\\\\\\hline e^{x}&+&+&+&+\\\\\\hline x-2&-&-&0&+\\\\\\hline x^{3}&-&+&+&+\\\\\\hline \\frac{e^{x}(x-2)}{x^{3}}&+&-&0&+\\\\\\hline \\end{array}$$" ] }, { "type": "step", "primary": "Identify the intervals that satisfy the required condition: $$>\\:0$$", "result": "x<0\\lor\\:x>2" } ], "meta": { "interimType": "Identify The Intervals NoCol 0Eq" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(x\\right)<0:{\\quad}0<x<2$$", "input": "\\frac{e^{x}\\left(x-2\\right)}{x^{3}}<0", "steps": [ { "type": "interim", "title": "Identify the intervals", "result": "0<x<2", "steps": [ { "type": "interim", "title": "Find the signs of the factors of $$\\frac{e^{x}\\left(x-2\\right)}{x^{3}}$$", "input": "\\frac{e^{x}\\left(x-2\\right)}{x^{3}}", "steps": [ { "type": "interim", "title": "Find the signs of $$e^{x}:{\\quad}$$Positive for all real values", "input": "e^{x}>0", "steps": [ { "type": "step", "primary": "If $$a>0,\\:a^{f\\left(x\\right)}\\:$$is greater than 0", "secondary": [ "$$a=e$$" ], "result": "\\mathrm{True\\:for\\:all}\\:x\\in\\mathbb{R}" } ], "meta": { "interimType": "Find Sign 1Eq" } }, { "type": "interim", "title": "Find the signs of $$x-2$$", "steps": [ { "type": "interim", "title": "$$x-2=0:{\\quad}x=2$$", 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Intervals NoCol 0Eq" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "Combine intervals with domain", "result": "-\\infty\\:<x<0,\\:0<x<2,\\:x=2,\\:2<x<\\infty\\:", "steps": [ { "type": "interim", "title": "Domain of $$\\frac{e^{x}}{x^{2}}\\::{\\quad}x<0\\lor\\:x>0$$", "steps": [ { "type": "definition", "title": "Domain definition", "text": "The domain of a function is the set of input or argument values for which the function is real and defined" }, { "type": "interim", "title": "Find undefined (singularity) points:$${\\quad}x=0$$", "input": "\\frac{e^{x}}{x^{2}}", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{e^{x}}{x^{2}}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$x^{2}=0:{\\quad}x=0$$", "input": "x^{2}=0", "steps": [ { "type": "step", "primary": "Apply rule $$x^n=0\\quad\\Rightarrow\\quad\\:x=0$$" }, { "type": "step", "result": "x=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The following points are undefined", "result": "x=0" } ], "meta": { "interimType": "Undefined Points 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6lOJ8p6oHaoHZ/UO5mSHLD1djVTVNfTJtGPIDKm6IXF/gNghAZYORfq9JtKwOORytPfV2oqc+75VRNLwQrPQk9pugt1g9hg3EUIob9qhnfZUgQUxJPyUNnGfVirkcwpVO92jAB5RE7BOdqi256y9oUI/L/NWLiz7WlXYJyi0meZnYoDSQN4diQ31DYw8UMxnuQ==" } }, { "type": "step", "primary": "The function domain", "result": "x<0\\lor\\:x>0" } ], "meta": { "solvingClass": "Function Domain", "interimType": "Function Domain Top 1Eq" } }, { "type": "interim", "title": "Combine $$x=0\\:$$ with domain:$${\\quad}$$False for all $$x\\in\\mathbb{R}$$", "input": "x=0\\land\\:\\left(x<0\\lor\\:x>0\\right)", "steps": [ { "type": "step", "primary": "Simplify", "result": "\\mathrm{False\\:for\\:all}\\:x\\in\\mathbb{R}" } ], "meta": { "interimType": "Combine With Domain 1Eq" } }, { "type": "interim", "title": "Combine $$x=2\\:$$ with domain:$${\\quad}x=2$$", "input": "x=2\\land\\:\\left(x<0\\lor\\:x>0\\right)", "steps": [ { "type": "step", "primary": "Simplify", "result": "x=2" } ], "meta": { "interimType": "Combine With Domain 1Eq" } }, { "type": "interim", "title": "Combine $$-\\infty\\:<x<0\\:$$ with domain:$${\\quad}-\\infty\\:<x<0$$", "input": "-\\infty\\:<x<0\\land\\:\\left(x<0\\lor\\:x>0\\right)", "steps": [ { "type": "step", "primary": "Simplify", "result": "-\\infty\\:<x<0" } ], "meta": { "interimType": "Combine With Domain 1Eq" } }, { "type": "interim", "title": "Combine $$2<x<\\infty\\:\\:$$ with domain:$${\\quad}2<x<\\infty\\:$$", "input": "2<x<\\infty\\:\\land\\:\\left(x<0\\lor\\:x>0\\right)", "steps": [ { "type": "step", "primary": "Simplify", "result": "2<x<\\infty\\:" } ], "meta": { "interimType": "Combine With Domain 1Eq" } }, { "type": "interim", "title": "Combine $$0<x<2\\:$$ with domain:$${\\quad}0<x<2$$", "input": 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"variable": "x", "plotRequest": "\\frac{e^{x}}{x^{2}}" }, "showViewLarger": true } } }