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: 11


$$\lim _{x \rightarrow-\frac{1}{2}} \frac{2 x+1}{2 x^{2}+3 x+1}=2$$

Work Step by Step

Given $$\lim _{x \rightarrow-\frac{1}{2}} \frac{2 x+1}{2 x^{2}+3 x+1}$$ let $$ f(x) = \frac{2 x+1}{2 x^{2}+3 x+1}$$ Since, we have $$ f(-\frac{1}{2})= \frac{-1+1}{\frac{1}{2}-\frac{3}{2}+1}=\frac{0}{0}$$ So, transform algebraically and cancel \begin{aligned} L&=\lim _{x \rightarrow-\frac{1}{2}} \frac{2 x+1}{2 x^{2}+3 x+1}\\ &=\lim _{x \rightarrow -\frac{1}{2}} \frac{(2x+1)}{(2x+1)(x+1)}\\ &=\lim _{x \rightarrow -\frac{1}{2}} \frac{1}{(x+1)}\\ &= \frac{1}{(-\frac{1}{2}+1)}\\ &= \frac{1}{(\frac{1}{2})}\\ &=2 \end{aligned}
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