Calculus (3rd Edition)

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

Chapter 2 - Limits - 2.6 Trigonometric Limits - Exercises - Page 77: 12


$$\lim _{x \rightarrow 0^{+}} \sqrt{x} 4^{\cos \left(\frac{\pi}{x}\right)}=0$$

Work Step by Step

For every $x \neq 0$, we have $\cos \frac{\pi}{x} \in[-1,1]$, which means that for every $x \neq 0$ we have $4^{\cos \frac{\pi}{x}} \in\left[4^{-1}, 4^{1}\right]=\left[\frac{1}{4}, 4\right]$. Therefore, we can put $l(x)=\frac{1}{4} \sqrt{x}$ and $u(x)=4 \sqrt{x},$ which gives us $ l(x) \leq f(x) \leq u(x) $ Then, by the Squeezing Theorem we have: $$\lim _{x \rightarrow 0^{+}} \sqrt{x} 4^{\cos \left(\frac{\pi}{x}\right)}=0$$
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