### 5.13. Calculate

$$
\lim _{x \rightarrow \pi}(\pi-x) \cot x
$$
Solution. The given limit represents an indeterminate form of type «0 $\cdot \infty »$. To resolve this, we introduce a new variable $\pi-x=t$. Then

$\lim _{x \rightarrow \pi}(\pi-x) \operatorname{ctg} x=\lim _{t \rightarrow 0} t \operatorname{ctg}(\pi-t)=-\lim _{t \rightarrow 0} t \operatorname{ctg} t=-\lim _{t \rightarrow 0} \frac{t}{\operatorname{tg} t}=-1$.