## Calculus (3rd Edition)

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