limit as x approaches 0 of x^{15x}

{ "query": { "display": "$$\\lim_{x\\to\\:0}\\left(x^{15x}\\right)$$", "symbolab_question": "BIG_OPERATOR#\\lim _{x\\to 0}(x^{15x})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Limits", "subTopic": "SingleVar", "default": "\\mathrm{Does\\:not\\:exist}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\lim_{x\\to\\:0}\\left(x^{15x}\\right)=$$Does not exist", "input": "\\lim_{x\\to\\:0}\\left(x^{15x}\\right)", "steps": [ { "type": "step", "primary": "If $$\\lim_{x\\to{a-}}{f\\left(x\\right)}\\neq\\lim_{x\\to{a+}}{f\\left(x\\right)}$$ then the limit does not exist", "meta": { "practiceLink": "/practice/limits-practice?subTopic=Divergence", "practiceTopic": "Limit divergence" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:0+}\\left(x^{15x}\\right)=1$$", "input": "\\lim_{x\\to\\:0+}\\left(x^{15x}\\right)", "steps": [ { "type": "interim", "title": "Apply exponent rules", "input": "x^{15x}", "result": "=\\lim_{x\\to\\:0+}\\left(e^{15x\\ln\\left(x\\right)}\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{x}=e^{\\ln\\left(a^{x}\\right)}$$", "result": "=e^{\\ln\\left(x^{15x}\\right)}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply log rule: $$\\ln\\left(a^{x}\\right)=x\\cdot\\ln\\left(a\\right)$$", "result": "=e^{15x\\ln\\left(x\\right)}", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Apply Exp Rules 0Eq" } }, { "type": "interim", "title": "Apply the Limit Chain Rule:$${\\quad}1$$", "input": "\\lim_{x\\to\\:0+}\\left(e^{15x\\ln\\left(x\\right)}\\right)", "steps": [ { "type": "definition", "title": "Limit Chain Rule", "text": "if $$\\lim_{u\\:\\to\\:b}\\:f\\left(u\\right)=L,\\:$$and $$\\lim_{x\\:\\to\\:a}g\\left(x\\right)=b,\\:$$and $$f\\left(x\\right)\\:$$is continuous at $$x=b$$<br/>$$\\quad$$Then: $$\\lim_{x\\:\\to\\:a}\\:f\\left(g\\left(x\\right)\\right)=L$$", "secondary": [ "$$g\\left(x\\right)=15x\\ln\\left(x\\right),\\:f\\left(u\\right)=e^{u}$$" ] }, { "type": "interim", "title": "$$\\lim_{x\\to\\:0+}\\left(15x\\ln\\left(x\\right)\\right)=0$$", "input": "\\lim_{x\\to\\:0+}\\left(15x\\ln\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$", "result": "=15\\cdot\\:\\lim_{x\\to\\:0+}\\left(x\\ln\\left(x\\right)\\right)" }, { "type": "interim", "title": "Rewrite for L'Hopital", "input": "\\lim_{x\\to\\:0+}\\left(x\\ln\\left(x\\right)\\right)", "result": "=15\\cdot\\:\\lim_{x\\to\\:0+}\\left(\\frac{\\ln\\left(x\\right)}{\\frac{1}{x}}\\right)", "steps": [ { "type": "step", "primary": "Indeterminate form: $$\\infty\\cdot0$$<br/>Rewrite to accomodate for L'Hopital" }, { "type": "step", "primary": "Apply the following algebraic property$$:{\\quad}a\\cdot\\:b=\\frac{a}{\\frac{1}{b}},\\:b\\ne0$$<br/>$$\\ln\\left(x\\right)x=\\frac{\\ln\\left(x\\right)}{\\frac{1}{x}}$$" }, { "type": "step", "result": "=15\\cdot\\:\\lim_{x\\to\\:0+}\\left(\\frac{\\ln\\left(x\\right)}{\\frac{1}{x}}\\right)" } ], "meta": { "interimType": "Limit Prepare for Lhopital Top 0Eq" } }, { "type": "interim", "title": "Apply L'Hopital's Rule", "input": "\\lim_{x\\to\\:0+}\\left(\\frac{\\ln\\left(x\\right)}{\\frac{1}{x}}\\right)", "result": "=15\\cdot\\:\\lim_{x\\to\\:0+}\\left(\\frac{\\frac{1}{x}}{-\\frac{1}{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\\:0+}\\left(\\frac{\\left(\\ln\\left(x\\right)\\right)^{^{\\prime}}}{\\left(\\frac{1}{x}\\right)^{^{\\prime}}}\\right)", "steps": [ { "type": "interim", "title": "$$\\lim_{x\\to\\:0+}\\left(\\ln\\left(x\\right)\\right)=-\\infty\\:$$", "input": "\\lim_{x\\to\\:0+}\\left(\\ln\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common limit: $$\\lim_{x\\to\\:0+}\\left(\\ln\\left(x\\right)\\right)=-\\infty\\:$$", "result": "=-\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:0+}\\left(\\frac{1}{x}\\right)=\\infty\\:$$", "input": "\\lim_{x\\to\\:0+}\\left(\\frac{1}{x}\\right)", "steps": [ { "type": "step", "primary": "For $$x\\:$$approaching $$0\\:$$from the right$$,\\:x>0\\quad\\Rightarrow\\quad\\:x>0$$", "secondary": [ "The denominator is a positive quantity approaching 0 from the right" ], "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "primary": "Meets L'hopital condition of: $$\\frac{-\\infty\\:}{\\infty\\:}$$", "result": "=\\lim_{x\\to\\:0+}\\left(\\frac{\\left(\\ln\\left(x\\right)\\right)^{^{^{\\prime}}}}{\\left(\\frac{1}{x}\\right)^{^{^{\\prime}}}}\\right)" } ], "meta": { "interimType": "Lhopital Condition 1Eq" } }, { "type": "interim", "title": "$$\\left(\\ln\\left(x\\right)\\right)'=\\frac{1}{x}$$", "input": "\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\right)=\\frac{1}{x}$$", "result": "=\\frac{1}{x}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYhHxrkiFdmQgNsZN21633mEcjlLRK1jUV206qo4+vRN78rEus7TgCihQBF5omOFkJlc0OBMs8qTL4oWnxx62vyTW8+5UeFz69+ZJ0Ew9+o4ULG0tPbHCza6x+uS5A0UR/rCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "$$\\left(\\frac{1}{x}\\right)'=-\\frac{1}{x^{2}}$$", "input": "\\frac{d}{dx}\\left(\\frac{1}{x}\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\frac{1}{a}=a^{-1}$$", "result": "=\\frac{d}{dx}\\left(x^{-1}\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=-1\\cdot\\:x^{-1-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$-1\\cdot\\:x^{-1-1}:{\\quad}-\\frac{1}{x^{2}}$$", "input": "-1\\cdot\\:x^{-1-1}", "result": "=-\\frac{1}{x^{2}}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$-1-1=-2$$", "result": "=-1\\cdot\\:x^{-2}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$x^{-2}=\\frac{1}{x^{2}}$$" ], "result": "=-1\\cdot\\:\\frac{1}{x^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\frac{1}{x^{2}}=\\frac{1}{x^{2}}$$", "result": "=-\\frac{1}{x^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+AgNZzeIicTfbr51/JGq1KiJTJQkRRngZl07rbqjeC6jkVi15I8rBefLi4Iyt2wrT1r2iaXj0z6hiHFOta70Gf8//6/nV5O4fb8Xgwi7mapyhd7tjiG+GxQNxDvGkZUl/dfOQfCkILEECUyPIy9DrRF1+E4wvPRIGnJs5KwUnrw=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\lim_{x\\to\\:0+}\\left(\\frac{\\frac{1}{x}}{-\\frac{1}{x^{2}}}\\right)" } ], "meta": { "interimType": "Lhopital Theorem 0Eq", "practiceLink": "/practice/limits-practice#area=main&subtopic=L'Hopital%20Rule", "practiceTopic": "L'hopital Rule", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+jbUbsVfpMbXIJ0UHfO5sTUTGxIy8toJdhWzXUxd5woF2Whsb5eOAREZpELUqtGCmfdn5+9BbYjO6KEj0cfiAweNnQmZ4ALdlISYzFZE1i5paMSH3+Uvodu1GkIiZwFWGam+VI5jQB4vQNat8+WwLv1LKfx4L2R8C21YlLRI72gLcIRZbU4wqJYx+ihDt2P869GNTQKYzG1oONPhai8MUjzwM6UL4mii3IPNdUmMIgfwt9LEn7QCBUukJKctfSJKU0E78/KwglSJgszhEVpVeroI8QZ8vlaOnWbVpXtZDHc=" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{1}{x}}{-\\frac{1}{x^{2}}}:{\\quad}-x$$", "input": "\\frac{\\frac{1}{x}}{-\\frac{1}{x^{2}}}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{\\frac{1}{x}}{\\frac{1}{x^{2}}}" }, { "type": "step", "primary": "Divide fractions: $$\\frac{\\frac{a}{b}}{\\frac{c}{d}}=\\frac{a\\cdot\\:d}{b\\cdot\\:c}$$", "result": "=-\\frac{1\\cdot\\:x^{2}}{x\\cdot\\:1}" }, { "type": "step", "primary": "Refine", "result": "=-\\frac{x^{2}}{x}" }, { "type": "step", "primary": "Cancel the common factor: $$x$$", "result": "=-x" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=15\\cdot\\:\\lim_{x\\to\\:0+}\\left(-x\\right)" }, { "type": "step", "primary": "Plug in the value $$x=0$$", "result": "=15\\left(-0\\right)", "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": "=0", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{u\\to\\:0+}\\left(e^{u}\\right)=1$$", "input": "\\lim_{u\\to\\:0+}\\left(e^{u}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$u=0$$", "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": "step", "primary": "By the chain rule:", "result": "=1" } ], "meta": { "interimType": "Limit Chain Rule 0Eq", "practiceLink": "/practice/limits-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Limit Chain Rule" } }, { "type": "step", "result": "=1" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:0-}\\left(x^{15x}\\right)=$$Does not exist", "input": "\\lim_{x\\to\\:0-}\\left(x^{15x}\\right)", "steps": [ { "type": "step", "primary": "$$x^{15x}\\:$$undefined for $$x\\:$$approaching $$0\\:$$from the left", "result": "=\\mathrm{Does\\:not\\:exist}" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=\\mathrm{Does\\:not\\:exist}" } ], "meta": { "practiceLink": "/practice/limits-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Limit Chain Rule" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "yes" }, "showViewLarger": true } }, "meta": { "showVerify": true } }