Example 4. Find $\lim _{x \rightarrow 0} x \operatorname{ctg} \frac{x}{3}$.
Solution. When $x \rightarrow 0$, we get an indeterminate form of $0 \cdot \infty$. Rewriting the given function in another form and applying formula (1.59), we find

$$
\lim _{x \rightarrow 0} x \operatorname{ctg} \frac{x}{3}=\lim _{x \rightarrow 0} x \frac{\cos \frac{x}{3}}{\sin \frac{x}{3}}=\lim _{x \rightarrow 0} \frac{x}{\sin \frac{x}{3}} \cos \frac{x}{3}=
$$

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