Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 1 - Functions and Limits - 1.6 Calculating Limits Using the Limit Laws - 1.6 Exercises: 61

Answer

please see step-by-step

Work Step by Step

We solve this applying the Squeeze Theorem, If $f(x) \leq g(x) \leq h(x)$ when $x$ is near $a$ (except possibly at a) and $\displaystyle \lim_{x\rightarrow a}f(x)=\lim_{x\rightarrow a}h(x)=L$ then $\displaystyle \lim_{x\rightarrow a}g(x)=L$ ---------- Let $u(x)=0,\qquad v(x)=x^{2}$ for all x. Then, $u(x) \leq f(x) \leq v(x)$ for all $x$. Also, $\displaystyle \lim_{x\rightarrow 0}u(x)= \displaystyle \lim_{x\rightarrow 0}0=0$ $\displaystyle \lim_{x\rightarrow 0}v(x)=\lim_{x\rightarrow 0}x^{2}=0,$ So, by the Squeeze Theorem, $\displaystyle \lim_{x\rightarrow 0}f(x)=0$.
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