Condition of the problem

Calculate the limit of the function:

$\lim _{x \rightarrow-2} \frac{\operatorname{tg} \pi x}{x+2}$
## Solution

Substitution:

$x=y-2 \Rightarrow y=x+2$

$x \rightarrow-2 \Rightarrow y \rightarrow 0$

We get:

$\lim _{x \rightarrow-2} \frac{\tan \pi x}{x+2}=\lim _{y \rightarrow 0} \frac{\tan \pi(y-2)}{(y-2)+2}=$

$=\lim _{y \rightarrow 0} \frac{\tan(\pi y-2 \pi)}{y}=\lim _{y \rightarrow 0} \frac{\tan \pi y}{y}=$

Using the substitution of equivalent infinitesimals:

$\tan \pi y \sim \pi y_{, \text {as }} y \rightarrow 0(\pi y \rightarrow 0)$

We get:

$=\lim _{y \rightarrow 0} \frac{\pi y}{y}=\lim _{y \rightarrow 0} \frac{\pi}{1}=\pi$

## Problem Kuznetsov Limits 13-23