Example 5. Find $\lim _{x \rightarrow 0} \frac{\sin a x}{x}$.
Solution. Let $a x=\alpha$, from which $x=\frac{\alpha}{a}$. If $x \rightarrow 0$, then $\alpha \rightarrow 0$, therefore

$$
\lim _{x \rightarrow 0} \frac{\sin a x}{x}=\lim _{\alpha \rightarrow 0} \frac{\sin \alpha}{\frac{\alpha}{a}}=\lim _{\alpha \rightarrow 0} a \frac{\sin \alpha}{\alpha}=a \lim _{\alpha \rightarrow 0} \frac{\sin \alpha}{\alpha}=a \cdot 1=a
$$

Thus,

$$
\lim _{x \rightarrow 0} \frac{\sin a x}{x}=a \text {. }
$$

In particular, when $a=2$

$$
\lim _{x \rightarrow 0} \frac{\sin 2 x}{x}=2
$$

when $a=\frac{1}{3}$ we have

$$
\lim _{x \rightarrow 0} \frac{\sin \frac{x}{3}}{x}=\frac{1}{3}
$$