## Problem Statement

Calculate the limit of the function:

$\lim _{x \rightarrow 0} \sqrt{\operatorname{arctg} x \cdot \sin ^{2}\left(\frac{1}{x}\right)+5 \cos x}$
## Solution

Since $\sin ^{2}\left(\frac{1}{x}\right)_{\text { is bounded, and }} \operatorname{arctg} x \rightarrow 0$, as $x \rightarrow 0$, then

$$
\operatorname{arctg} x \cdot \sin ^{2}\left(\frac{1}{x}\right) \rightarrow 0 \underset{\text { as } x \rightarrow 0}{ }
$$

Then:

$\lim _{x \rightarrow 0} \sqrt{\operatorname{arctg} x \cdot \sin ^{2}\left(\frac{1}{x}\right)+5 \cos x}=\sqrt{0+5 \cos 0}=\sqrt{5 \cdot 1}=\sqrt{5}$