## Problem Statement

Calculate the limit of the function:

$$
\lim _{x \rightarrow 0}\left(\frac{\operatorname{arctg} 3 x}{x}\right)^{x+2}
$$
## Solution

$$
\begin{aligned}
& \lim _{x \rightarrow 0}\left(\frac{\operatorname{arctg} 3 x}{x}\right)^{x+2}=\left(\lim _{x \rightarrow 0} \frac{\operatorname{arctg} 3 x}{x}\right)^{\lim _{x \rightarrow 0} x+2}= \\
& =\left(\lim _{x \rightarrow 0} \frac{\operatorname{arctg} 3 x}{x}\right)^{0+2}=\left(\lim _{x \rightarrow 0} \frac{\operatorname{arctg} 3 x}{x}\right)^{2}=
\end{aligned}
$$

$\operatorname{arctg} 3 x \sim 3 x$, as $x \rightarrow 0(3 x \rightarrow 0)$

We get:

$=\left(\lim _{x \rightarrow 0} \frac{3 x}{x}\right)^{2}=\left(\lim _{x \rightarrow 0} 3\right)^{2}=3^{2}=9$

## Problem Kuznetsov Limits 18-24