Question

$\underset{x\to \infty }{\lim }\frac{\sin x}{x}$ equals

(a) 0
(b) ∞
(c) 1
(d) does not exist

Answer 1

(a)
$$\begin{gathered} \underset{x\to \infty }{\lim }\frac{\sin x}{x} \\ \text{L}\mathrm{et}x=\frac{1}{y} \\ x\to \infty \\ \therefore y\to 0 \\ =\underset{y\to 0}{\lim }\frac{\sin {\displaystyle \frac{1}{y}}}{{\displaystyle \frac{1}{y}}} \\ =\underset{y\to 0}{\lim }y\sin \frac{1}{y} \\ =\underset{y\to 0}{\lim }y\times \underset{y\to 0}{\lim }\sin \frac{1}{y} \\ =0\times \underset{y\to 0}{\lim }\sin \frac{1}{y} \\ =0 \end{gathered}$$