Calculus 8th Edition

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

Chapter 6 - Inverse Functions - 6.2 Exponential Functions and Their Derivatives - 6.2 Exercises - Page 418: 29



Work Step by Step

Need to find the limit for $\lim\limits_{x \to \infty} (e^{-2x}cosx)$. $\lim\limits_{x \to \infty}(e^{-2x}cosx)=\lim\limits_{x \to \infty}\frac{cosx}{e^{2x}}$ The limit for $cosx$ lies between -1 to +1 any finite value. Therefore, $\lim\limits_{x \to \infty}\frac{cosx}{e^{2x}}=\frac{\lim\limits_{x \to \infty}cosx}{\lim\limits_{x \to \infty}e^{2x}}$ since $-1\leq cosx\leq1$ Hence, $\lim\limits_{x \to \infty}(e^{-2x}cosx)$=finite value divided by $\infty$ approaches to $0$.
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