limit as x approaches-1 of (x^3)/(x^2+1)

{ "query": { "display": "$$\\lim_{x\\to\\:-1}\\left(\\frac{x^{3}}{x^{2}+1}\\right)$$", "symbolab_question": "BIG_OPERATOR#\\lim _{x\\to -1}(\\frac{x^{3}}{x^{2}+1})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Limits", "subTopic": "SingleVar", "default": "-\\frac{1}{2}", "decimal": "-0.5", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\lim_{x\\to\\:-1}\\left(\\frac{x^{3}}{x^{2}+1}\\right)=-\\frac{1}{2}$$", "input": "\\lim_{x\\to\\:-1}\\left(\\frac{x^{3}}{x^{2}+1}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=-1$$", "result": "=\\frac{\\left(-1\\right)^{3}}{\\left(-1\\right)^{2}+1}", "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 $$\\frac{\\left(-1\\right)^{3}}{\\left(-1\\right)^{2}+1}:{\\quad}-\\frac{1}{2}$$", "input": "\\frac{\\left(-1\\right)^{3}}{\\left(-1\\right)^{2}+1}", "result": "=-\\frac{1}{2}", "steps": [ { "type": "interim", "title": "$$\\left(-1\\right)^{3}=-1$$", "input": "\\left(-1\\right)^{3}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=-a^{n},\\:$$if $$n$$ is odd", "secondary": [ "$$\\left(-1\\right)^{3}=-1^{3}$$" ], "result": "=-1^{3}" }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ClCeQdnpyQa/XVdLG7jeqM0ag8T1MwTer44+aCS/ZFDVPYg32V0aoZlEjRGoE3SC92wC37GgJ7iWYZErnwi6GR47EHm2iOOTglAC5fu7cyE=" } }, { "type": "step", "result": "=\\frac{-1}{\\left(-1\\right)^{2}+1}" }, { "type": "interim", "title": "$$\\left(-1\\right)^{2}+1=2$$", "input": "\\left(-1\\right)^{2}+1", "steps": [ { "type": "interim", "title": "$$\\left(-1\\right)^{2}=1$$", "input": "\\left(-1\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-1\\right)^{2}=1^{2}$$" ], "result": "=1^{2}" }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78E1FVQW6YvXK7raPRxih+c0ag8T1MwTer44+aCS/ZFAdx7pcd1x/bAWpIL8hAintf05A2GsVmPba4FjoW22b4iKyMg44e9p5G7GRfJ2en9g=" } }, { "type": "step", "result": "=1+1" }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s75ORzeCgnWl7uAJU70MaKjCAn9lkDfZkicUGkO3EF+IpFwBI6pGksh4jJQHOAL6dQv3o04UaokLQVdkFqWWrZq2q4jhJbmmkwLgnnNEW42Vw=" } }, { "type": "step", "result": "=\\frac{-1}{2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{1}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7G18XghIrwPPl4902UO6BvP3pzF3EE8eK/Tc9Tx5aLUBV00rpv8+ZC6TM10tVCSHshgFkMSyGhkcJufVqsMeq9A4Wg7zh4ZA/tWZIC5aAUJN6pfF1z6umzUJTJvt+ojYZY4LVEbDWfJhdqtpUA6/Oh5frovOXpNVFe/d814gla7cx9c893bdFcCbFWky7MB/C" } } ], "meta": { "solvingClass": "Limits", "practiceLink": "/practice/limits-practice", "practiceTopic": "Limits" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "yes" }, "showViewLarger": true } }, "meta": { "showVerify": true } }