Calculus (3rd Edition)

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

Chapter 2 - Limits - 2.7 Limits at Infinity - Preliminary Questions - Page 82: 6


$$\lim _{x \rightarrow \infty}\sin \frac{1}{x}=0 .$$

Work Step by Step

When $ x\to \infty $ the function $\frac{1}{x}\to 0$ and hence $\sin \frac{1}{x}\to \sin 0=0$. But when $ x\to 0$ the function $\frac{1}{x}\to \infty $ and hence $\sin \frac{1}{x}\to \sin \infty $ and the function $\sin x $ oscillates indefinitely. Thus, we have $$\lim _{x \rightarrow \infty}\sin \frac{1}{x}=0 .$$
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