limit as x approaches-1 of (3x-7)/(2x+5)

{ "query": { "display": "$$\\lim_{x\\to\\:-1}\\left(\\frac{3x-7}{2x+5}\\right)$$", "symbolab_question": "BIG_OPERATOR#\\lim _{x\\to -1}(\\frac{3x-7}{2x+5})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Limits", "subTopic": "SingleVar", "default": "-\\frac{10}{3}", "decimal": "-3.33333…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\lim_{x\\to\\:-1}\\left(\\frac{3x-7}{2x+5}\\right)=-\\frac{10}{3}$$", "input": "\\lim_{x\\to\\:-1}\\left(\\frac{3x-7}{2x+5}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=-1$$", "result": "=\\frac{3\\left(-1\\right)-7}{2\\left(-1\\right)+5}", "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 $$\\frac{3\\left(-1\\right)-7}{2\\left(-1\\right)+5}:{\\quad}-\\frac{10}{3}$$", "input": "\\frac{3\\left(-1\\right)-7}{2\\left(-1\\right)+5}", "result": "=-\\frac{10}{3}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-3\\cdot\\:1-7}{-2\\cdot\\:1+5}" }, { "type": "interim", "title": "$$-3\\cdot\\:1-7=-10$$", "input": "-3\\cdot\\:1-7", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:1=3$$", "result": "=-3-7" }, { "type": "step", "primary": "Subtract the numbers: $$-3-7=-10$$", "result": "=-10" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ciuQxNMCFZDBI4/VtBycGlXTSum/z5kLpMzXS1UJIewogZYyP6viKuQt0bEoFXamuDTa1v+ZaPma7/p91PqrJeTzvmpZEWHtgi44/UxrSAkkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=\\frac{-10}{-2\\cdot\\:1+5}" }, { "type": "interim", "title": "$$-2\\cdot\\:1+5=3$$", "input": "-2\\cdot\\:1+5", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=-2+5" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-2+5=3$$", "result": "=3" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7oZMfcTr0ZbQNIiY6I7VtlVXTSum/z5kLpMzXS1UJIex+QJ/6cxbtT+U96luRwKKo0xRQi4z1QuU1Jwaqk+96tk9TAMjOaHnyEvDiHvvEHZc=" } }, { "type": "step", "result": "=\\frac{-10}{3}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{10}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ktvPgCpcn6AP3f5PNHIlBX+7RjBuHdQpWJbOTaM5gI/NGoPE9TME3q+OPmgkv2RQrRKMQ0pA8A67jlmbMMkBF7ImM7O02I4xVN41NJp/Y96LGmNnLPWGf9PH3lpmjoJIKZsOh7prTrQXWQKxZS7dEJMCkKHAV7pQAX5GG7yt3EewiNrEngO+NNvZ9sqNu+2V" } } ], "meta": { "solvingClass": "Limits", "practiceLink": "/practice/limits-practice", "practiceTopic": "Limits" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "yes" }, "showViewLarger": true } }, "meta": { "showVerify": true } }