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 - Page 71: 37



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 $f(x)$ is "squeezed" between$ u(x)=4x-9 $ and $v(x)=x^{2}-4x+7$ $u(x) \leq f(x) \leq v(x)$, for $x\geq 0$. $\displaystyle \lim_{x\rightarrow 4}u(x)= \displaystyle \lim_{x\rightarrow 4}(4x-9)=4(4)-9=7$ $\displaystyle \lim_{x\rightarrow 4} v(x) \displaystyle \lim_{x\rightarrow 4}(x^{2}-4x+7)=4^{2}-4(4)+7=7$. By the Squeeze Theorem, $\displaystyle \lim_{x\rightarrow 4}f(x)=7$
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