limit as y approaches 6 of (2y^2-15y+18)/(3y^2-17y-6)

{ "query": { "display": "$$\\lim_{y\\to\\:6}\\left(\\frac{2y^{2}-15y+18}{3y^{2}-17y-6}\\right)$$", "symbolab_question": "BIG_OPERATOR#\\lim _{y\\to 6}(\\frac{2y^{2}-15y+18}{3y^{2}-17y-6})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Limits", "subTopic": "SingleVar", "default": "\\frac{9}{19}", "decimal": "0.47368…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\lim_{y\\to\\:6}\\left(\\frac{2y^{2}-15y+18}{3y^{2}-17y-6}\\right)=\\frac{9}{19}$$", "input": "\\lim_{y\\to\\:6}\\left(\\frac{2y^{2}-15y+18}{3y^{2}-17y-6}\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2y^{2}-15y+18}{3y^{2}-17y-6}:{\\quad}\\frac{2y-3}{3y+1}$$", "input": "\\frac{2y^{2}-15y+18}{3y^{2}-17y-6}", "steps": [ { "type": "interim", "title": "Factor $$2y^{2}-15y+18:{\\quad}\\left(2y-3\\right)\\left(y-6\\right)$$", "input": "2y^{2}-15y+18", "result": "=\\frac{\\left(2y-3\\right)\\left(y-6\\right)}{3y^{2}-17y-6}", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "2y^{2}-15y+18", "steps": [ { "type": "definition", "title": "Definition", "text": "For $$ax^{2}+bx+c\\:$$find $$u,\\:v\\:$$ such that: $$u\\cdot\\:v=a\\cdot\\:c\\:$$and $$u+v=b$$<br/>and group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "secondary": [ "$$a=2,\\:b=-15,\\:c=18$$", "$$u*v=36,\\:u+v=-15$$" ] }, { "type": "interim", "title": "Factors of $$36:{\\quad}1,\\:2,\\:3,\\:4,\\:6,\\:9,\\:12,\\:18,\\:36$$", "input": "36", "steps": [ { "type": "definition", "title": "Divisors (Factors)", "text": "Factors are numbers we can multiply together to get another number" }, { "type": "interim", "title": "Find the Prime factors of $$36:{\\quad}2,\\:2,\\:3,\\:3$$", "input": "36", "steps": [ { "type": "step", "primary": "$$36\\:$$divides by $$2\\quad\\:36=18\\cdot\\:2$$", "result": "=2\\cdot\\:18" }, { "type": "step", "primary": "$$18\\:$$divides by $$2\\quad\\:18=9\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:9" }, { "type": "step", "primary": "$$9\\:$$divides by $$3\\quad\\:9=3\\cdot\\:3$$", "result": "=2\\cdot\\:2\\cdot\\:3\\cdot\\:3" }, { "type": "step", "primary": "$$2,\\:3$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:2\\cdot\\:3\\cdot\\:3" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRhCh5B6yDFXoxwojiAQa7+iBBTEk/JQ2cZ9WKuRzClU7UyQJGJhlEk8mqOp0dvktKhuQbtB5yLYoe1JMepWppwNsI5B83GgK4Rq+y4sowjuo" } }, { "type": "interim", "title": "Multiply the prime factors of $$36:{\\quad}4,\\:6,\\:12,\\:9,\\:18$$", "result": "4,\\:6,\\:12,\\:9,\\:18", "steps": [ { "type": "step", "primary": "$$2\\cdot\\:2=4$$", "secondary": [ "$$2\\cdot\\:3=6$$", "$$2\\cdot\\:2\\cdot\\:3=12$$", "$$3\\cdot\\:3=9$$", "$$2\\cdot\\:3\\cdot\\:3=18$$" ] }, { "type": "step", "result": "4,\\:6,\\:12,\\:9,\\:18" } ], "meta": { "interimType": "Multiply the prime factors 1Eq" } }, { "type": "step", "primary": "Add the prime factors: ", "result": "2,\\:3" }, { "type": "step", "primary": "Add 1 and the number $$36\\:$$ itself", "result": "1,\\:36" }, { "type": "step", "primary": "The factors of $$36$$", "result": "1,\\:2,\\:3,\\:4,\\:6,\\:9,\\:12,\\:18,\\:36" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Negative factors of $$36:{\\quad}-1,\\:-2,\\:-3,\\:-4,\\:-6,\\:-9,\\:-12,\\:-18,\\:-36$$", "steps": [ { "type": "step", "primary": "Multiply the factors by $$-1$$ to get the negative factors", "result": "-1,\\:-2,\\:-3,\\:-4,\\:-6,\\:-9,\\:-12,\\:-18,\\:-36" } ], "meta": { "interimType": "Negative Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=36,\\:$$check if $$u+v=-15$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=36:\\quad\\:u*v=36,\\:u+v=37\\quad\\Rightarrow\\quad\\:$$False", "secondary": [ "Check $$u=2,\\:v=18:\\quad\\:u*v=36,\\:u+v=20\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=3,\\:v=12:\\quad\\:u*v=36,\\:u+v=15\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=4,\\:v=9:\\quad\\:u*v=36,\\:u+v=13\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=6,\\:v=6:\\quad\\:u*v=36,\\:u+v=12\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=-1,\\:v=-36:\\quad\\:u*v=36,\\:u+v=-37\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=-2,\\:v=-18:\\quad\\:u*v=36,\\:u+v=-20\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=-3,\\:v=-12:\\quad\\:u*v=36,\\:u+v=-15\\quad\\Rightarrow\\quad\\:$$True", "Check $$u=-4,\\:v=-9:\\quad\\:u*v=36,\\:u+v=-13\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=-6,\\:v=-6:\\quad\\:u*v=36,\\:u+v=-12\\quad\\Rightarrow\\quad\\:$$False" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=-3,\\:v=-12" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(2y^{2}-3y\\right)+\\left(-12y+18\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMws0s2e4LypmPzpvXw67IIqWk3hxk9aCfAWodBRxXgUexnyLfTdVp2W4I+K/E5MHE+hcj6isJZHmVLbwm+mZ9aY4/ML/bQJqqxyc/C+D4Qny1hInPPKIict/KnzARdgy3GPC30sSftAIFS6Qkpy19IkpO9G97ntiv4zbNidgqi4bUzpTv/NtfYqJeFljyKdNL9y+TbJ6Ohy9ptAHnj8WUMjY=" } }, { "type": "step", "result": "=\\left(2y^{2}-3y\\right)+\\left(-12y+18\\right)" }, { "type": "interim", "title": "Factor out $$y\\:$$from $$2y^{2}-3y:\\quad\\:y\\left(2y-3\\right)$$", "input": "2y^{2}-3y", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$y^{2}=yy$$" ], "result": "=2yy-3y", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$y$$", "result": "=y\\left(2y-3\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "interim", "title": "Factor out $$-6\\:$$from $$-12y+18:\\quad\\:-6\\left(2y-3\\right)$$", "input": "-12y+18", "steps": [ { "type": "step", "primary": "Rewrite $$18$$ as $$6\\cdot\\:3$$", "secondary": [ "Rewrite $$12$$ as $$6\\cdot\\:2$$" ], "result": "=-6\\cdot\\:2y+6\\cdot\\:3" }, { "type": "step", "primary": "Factor out common term $$-6$$", "result": "=-6\\left(2y-3\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "step", "result": "=y\\left(2y-3\\right)-6\\left(2y-3\\right)" }, { "type": "step", "primary": "Factor out common term $$2y-3$$", "result": "=\\left(2y-3\\right)\\left(y-6\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Factor $$3y^{2}-17y-6:{\\quad}\\left(3y+1\\right)\\left(y-6\\right)$$", "input": "3y^{2}-17y-6", "result": "=\\frac{\\left(2y-3\\right)\\left(y-6\\right)}{\\left(3y+1\\right)\\left(y-6\\right)}", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "3y^{2}-17y-6", "steps": [ { "type": "definition", "title": "Definition", "text": "For $$ax^{2}+bx+c\\:$$find $$u,\\:v\\:$$ such that: $$u\\cdot\\:v=a\\cdot\\:c\\:$$and $$u+v=b$$<br/>and group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "secondary": [ "$$a=3,\\:b=-17,\\:c=-6$$", "$$u*v=-18,\\:u+v=-17$$" ] }, { "type": "interim", "title": "Factors of $$18:{\\quad}1,\\:2,\\:3,\\:6,\\:9,\\:18$$", "input": "18", "steps": [ { "type": "definition", "title": "Divisors (Factors)", "text": "Factors are numbers we can multiply together to get another number" }, { "type": "interim", "title": "Find the Prime factors of $$18:{\\quad}2,\\:3,\\:3$$", "input": "18", "steps": [ { "type": "step", "primary": "$$18\\:$$divides by $$2\\quad\\:18=9\\cdot\\:2$$", "result": "=2\\cdot\\:9" }, { "type": "step", "primary": "$$9\\:$$divides by $$3\\quad\\:9=3\\cdot\\:3$$", "result": "=2\\cdot\\:3\\cdot\\:3" }, { "type": "step", "primary": "$$2,\\:3$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:3\\cdot\\:3" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRpR1IIr0M1tj/hG6BCqscuWBBTEk/JQ2cZ9WKuRzClU7UyQJGJhlEk8mqOp0dvktKkavJ+4fkE2eZqED6K/FpGXooWAuAHKtvQokwVYAX2ga" } }, { "type": "interim", "title": "Multiply the prime factors of $$18:{\\quad}6,\\:9$$", "result": "6,\\:9", "steps": [ { "type": "step", "primary": "$$2\\cdot\\:3=6$$", "secondary": [ "$$3\\cdot\\:3=9$$" ] }, { "type": "step", "result": "6,\\:9" } ], "meta": { "interimType": "Multiply the prime factors 1Eq" } }, { "type": "step", "primary": "Add the prime factors: ", "result": "2,\\:3" }, { "type": "step", "primary": "Add 1 and the number $$18\\:$$ itself", "result": "1,\\:18" }, { "type": "step", "primary": "The factors of $$18$$", "result": "1,\\:2,\\:3,\\:6,\\:9,\\:18" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Negative factors of $$18:{\\quad}-1,\\:-2,\\:-3,\\:-6,\\:-9,\\:-18$$", "steps": [ { "type": "step", "primary": "Multiply the factors by $$-1$$ to get the negative factors", "result": "-1,\\:-2,\\:-3,\\:-6,\\:-9,\\:-18" } ], "meta": { "interimType": "Negative Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=-18,\\:$$check if $$u+v=-17$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=-18:\\quad\\:u*v=-18,\\:u+v=-17\\quad\\Rightarrow\\quad\\:$$True", "secondary": [ "Check $$u=2,\\:v=-9:\\quad\\:u*v=-18,\\:u+v=-7\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=3,\\:v=-6:\\quad\\:u*v=-18,\\:u+v=-3\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=6,\\:v=-3:\\quad\\:u*v=-18,\\:u+v=3\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=9,\\:v=-2:\\quad\\:u*v=-18,\\:u+v=7\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=18,\\:v=-1:\\quad\\:u*v=-18,\\:u+v=17\\quad\\Rightarrow\\quad\\:$$False" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=1,\\:v=-18" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(3y^{2}+y\\right)+\\left(-18y-6\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwtAaM5hPtVxPJo2BvslcxGpJQz2lqSQogu9PoWz88zfnRIVQdYp0ZW/Nzw1XCNMUXh4Aty1i0DpNCDInxApaAMMg9FboRAWKm1lIrj8WX819Aa8JwFfCzqv/bcTIoBfNmU3kCh3oevUunZ7/b0qFKBSWsWV28wexhpgboh1m5Hi0Fimz7IJvR01OX8go+hj05ompXFf3SOUx+H18qfp3MLg=" } }, { "type": "step", "result": "=\\left(3y^{2}+y\\right)+\\left(-18y-6\\right)" }, { "type": "interim", "title": "Factor out $$y\\:$$from $$3y^{2}+y:\\quad\\:y\\left(3y+1\\right)$$", "input": "3y^{2}+y", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$y^{2}=yy$$" ], "result": "=3yy+y", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$y$$", "result": "=y\\left(3y+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "interim", "title": "Factor out $$-6\\:$$from $$-18y-6:\\quad\\:-6\\left(3y+1\\right)$$", "input": "-18y-6", "steps": [ { "type": "step", "primary": "Rewrite $$18$$ as $$6\\cdot\\:3$$", "result": "=-6\\cdot\\:3y-6" }, { "type": "step", "primary": "Factor out common term $$-6$$", "result": "=-6\\left(3y+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "step", "result": "=y\\left(3y+1\\right)-6\\left(3y+1\\right)" }, { "type": "step", "primary": "Factor out common term $$3y+1$$", "result": "=\\left(3y+1\\right)\\left(y-6\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$y-6$$", "result": "=\\frac{2y-3}{3y+1}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\lim_{y\\to\\:6}\\left(\\frac{2y-3}{3y+1}\\right)" }, { "type": "step", "primary": "Plug in the value $$y=6$$", "result": "=\\frac{2\\cdot\\:6-3}{3\\cdot\\:6+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": "step", "primary": "Simplify", "result": "=\\frac{9}{19}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "practiceLink": "/practice/limits-practice", "practiceTopic": "Limits" } }, "plot_output": { "meta": { "plotInfo": { "variable": "y", "plotRequest": "yes" }, "showViewLarger": true } }, "meta": { "showVerify": true } }