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Calculus Level 2

Find the value of lim ⁡ y → 2 y 3 − y 2 − y − 2 2 y 3 − 5 y 2 + 5 y − 6 \large{\displaystyle{\lim_{y\rightarrow2}\dfrac{y^3-y^2-y-2}{2y^3-5y^2+5y-6}}} .

9 5 \dfrac{9}{5} 7 9 \dfrac{7}{9} 11 17 \dfrac{11}{17} 0 0

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1 solution

Relevant wiki: L'Hopital's Rule - Basic

Since by substitution gives 0 0 \dfrac{0}{0} , we use the L'Hopital's Rule .

lim ⁡ y → 2 y 3 − y 2 − y − 2 2 y 3 − 5 y 2 + 5 y − 6 = lim ⁡ y → 2 3 y 2 − 2 y − 1 6 y 2 − 10 y + 5 = 3 ( 2 2 ) − 2 ( 2 ) − 1 6 ( 2 2 ) − 10 ( 2 ) + 5 = 12 − 4 − 1 24 − 20 + 5 = \displaystyle{\lim_{y\rightarrow2}\dfrac{y^3-y^2-y-2}{2y^3-5y^2+5y-6}}=\displaystyle{\lim_{y\rightarrow2}\dfrac{3y^2-2y-1}{6y^2-10y+5}}=\dfrac{3(2^2)-2(2)-1}{6(2^2)-10(2)+5}=\dfrac{12-4-1}{24-20+5}= 7 9 \boxed{\dfrac{7}{9}}

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