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: 38



Work Step by Step

Theorem 3 - 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$\quad \displaystyle \lim_{x\rightarrow a}g(x)=L$ -------------- Our $g(x)$ is "squeezed" between$ u(x)=2x $ and $v(x)=x^{4}-x^{2}+2$ $u(x) \leq g(x) \leq v(x)$, for all x.. $\displaystyle \lim_{x\rightarrow 1}u(x)= \displaystyle \lim_{x\rightarrow 1}(2x)=2(1)=2$ $\displaystyle \lim_{x\rightarrow 1} v(x)=\displaystyle \lim_{x\rightarrow 1}(x^{4}-x^{2}+2)=1^{4}-1^{2}+2=2$. By the Squeeze Theorem, $\displaystyle \lim_{x\rightarrow 1}g(x)=2$
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