Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 2 - Limits - 2.5 Evaluating Limits Algebraically - Exercises - Page 72: 20


$$\lim _{y \rightarrow 3} \frac{y^{2}+y-12}{y^{3}-10 y+3} =\frac{7}{17}$$

Work Step by Step

Given $$\lim _{y \rightarrow 3} \frac{y^{2}+y-12}{y^{3}-10 y+3}$$ let $$ f(y) = \frac{y^{2}+y-12}{y^{3}-10 y+3}$$ Since, we have $$ f(3)= \frac{9+3-12}{27-30+3}=\frac{0}{0}$$ So, transform algebraically and cancel \begin{aligned}L&=\lim _{y \rightarrow 3} \frac{y^{2}+y-12}{y^{3}-10 y+3}\\ &=\lim _{y \rightarrow 3} \frac{(y-3)(y+4)}{(y-3)\left(y^{2}+3 y-1\right)}\\ &=\lim _{y \rightarrow 3} \frac{(y+4)}{\left(y^{2}+3 y-1\right)}\\ &=\frac{3+4}{9+9-1}\\ &=\frac{7}{17} \end{aligned}
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