## Problem Statement

Calculate the limit of the function:

$\lim _{x \rightarrow 0} \sqrt{2 \cos ^{2} x+\left(e^{x}-1\right) \sin \left(\frac{1}{x}\right)}$
Solution

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

Then:

$$
\begin{aligned}
& \lim _{x \rightarrow 0} \sqrt{2 \cos ^{2} x+\left(e^{x}-1\right) \sin \left(\frac{1}{x}\right)}=\sqrt{2 \cos ^{2} 0+0}= \\
& =\sqrt{2 \cdot 1^{2}+0}=\sqrt{2}
\end{aligned}
$$