limit as x approaches 1/4 of 8x(x-1/5)

{ "query": { "display": "$$\\lim_{x\\to\\:\\frac{1}{4}}\\left(8x\\left(x-\\frac{1}{5}\\right)\\right)$$", "symbolab_question": "BIG_OPERATOR#\\lim _{x\\to \\frac{1}{4}}(8x(x-\\frac{1}{5}))" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Limits", "subTopic": "SingleVar", "default": "\\frac{1}{10}", "decimal": "0.1", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\lim_{x\\to\\:\\frac{1}{4}}\\left(8x\\left(x-\\frac{1}{5}\\right)\\right)=\\frac{1}{10}$$", "input": "\\lim_{x\\to\\:\\frac{1}{4}}\\left(8x\\left(x-\\frac{1}{5}\\right)\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=\\frac{1}{4}$$", "result": "=8\\cdot\\:\\frac{1}{4}\\left(\\frac{1}{4}-\\frac{1}{5}\\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": "interim", "title": "Simplify $$8\\cdot\\:\\frac{1}{4}\\left(\\frac{1}{4}-\\frac{1}{5}\\right):{\\quad}\\frac{1}{10}$$", "input": "8\\cdot\\:\\frac{1}{4}\\left(\\frac{1}{4}-\\frac{1}{5}\\right)", "result": "=\\frac{1}{10}", "steps": [ { "type": "interim", "title": "Join $$\\frac{1}{4}-\\frac{1}{5}:{\\quad}\\frac{1}{20}$$", "input": "\\frac{1}{4}-\\frac{1}{5}", "result": "=8\\cdot\\:\\frac{1}{4}\\cdot\\:\\frac{1}{20}", "steps": [ { "type": "interim", "title": "Least Common Multiplier of $$4,\\:5:{\\quad}20$$", "input": "4,\\:5", "steps": [ { "type": "definition", "title": "Least Common Multiplier (LCM)", "text": "The LCM of $$a,\\:b$$ is the smallest positive number that is divisible by both $$a$$ and $$b$$" }, { "type": "interim", "title": "Prime factorization of $$4:{\\quad}2\\cdot\\:2$$", "input": "4", "steps": [ { "type": "step", "primary": "$$4\\:$$divides by $$2\\quad\\:4=2\\cdot\\:2$$", "result": "=2\\cdot\\:2" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRsG/uC0ndYtZpJL4uAxK7FI/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp39fF/zAtU5baHQ1hwgXA+n" } }, { "type": "interim", "title": "Prime factorization of $$5:{\\quad}5$$", "input": "5", "steps": [ { "type": "step", "primary": "$$5$$ is a prime number, therefore no factorization is possible", "result": "=5" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRjl/dE9e0owjU0NK6lxSAv4/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp3mWpvzkJh0pk9SzVPr3Sj8" } }, { "type": "step", "primary": "Multiply each factor the greatest number of times it occurs in either $$4$$ or $$5$$", "result": "=2\\cdot\\:2\\cdot\\:5" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2\\cdot\\:5=20$$", "result": "=20" } ], "meta": { "solvingClass": "LCM", "interimType": "LCM Top 1Eq" } }, { "type": "interim", "title": "Adjust Fractions based on the LCM", "steps": [ { "type": "step", "primary": "Multiply each numerator by the same amount needed to multiply its<br/>corresponding denominator to turn it into the LCM $$20$$" }, { "type": "step", "primary": "For $$\\frac{1}{4}:\\:$$multiply the denominator and numerator by $$5$$", "result": "\\frac{1}{4}=\\frac{1\\cdot\\:5}{4\\cdot\\:5}=\\frac{5}{20}" }, { "type": "step", "primary": "For $$\\frac{1}{5}:\\:$$multiply the denominator and numerator by $$4$$", "result": "\\frac{1}{5}=\\frac{1\\cdot\\:4}{5\\cdot\\:4}=\\frac{4}{20}" } ], "meta": { "interimType": "LCD Adjust Fractions 1Eq" } }, { "type": "step", "result": "=\\frac{5}{20}-\\frac{4}{20}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{5-4}{20}" }, { "type": "step", "primary": "Subtract the numbers: $$5-4=1$$", "result": "=\\frac{1}{20}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}\\cdot\\frac{d}{e}=\\frac{a\\:\\cdot\\:b\\:\\cdot\\:d}{c\\:\\cdot\\:e}$$", "result": "=\\frac{1\\cdot\\:1\\cdot\\:8}{4\\cdot\\:20}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1\\cdot\\:8=8$$", "result": "=\\frac{8}{4\\cdot\\:20}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:20=80$$", "result": "=\\frac{8}{80}" }, { "type": "step", "primary": "Cancel the common factor: $$8$$", "result": "=\\frac{1}{10}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s77UAJCwYWq4vVRQWGBsw5iprJNWg++Yt4D2WTW2Mt7jpoPmyWdwwHhwL8SGD8Vu6X0y8fJjnR2Fc8V+uNyaihSELJUwysTkf+ifmThr1HGrrQr59mWfLlocfUs9gCymHigQUxJPyUNnGfVirkcwpVO7NcHgE3NWLVU0b18yzRWa4kJNp7gjxf6BoYVjNDOGvaGQeRSa2/KZ1ktl9shR1JScJKnn43tZl/5r4OQELi0ig=" } } ], "meta": { "solvingClass": "Limits", "practiceLink": "/practice/limits-practice", "practiceTopic": "Limits" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "yes" }, "showViewLarger": true } }, "meta": { "showVerify": true } }