## Problem Statement

Calculate the limit of the function:

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

Since $\sqrt{2+\cos \left(\frac{1}{x}\right)}$ is bounded, and $\ln (1+x) \rightarrow 0$, as $x \rightarrow 0$, then $\ln (1+x) \sqrt{2+\cos \left(\frac{1}{x}\right)} \rightarrow 0 \quad$, as $x \rightarrow 0$

Then:

$\lim _{x \rightarrow 0} \frac{\cos x+\ln (1+x) \sqrt{2+\cos \left(\frac{1}{x}\right)}}{2+e^{x}}=\frac{\cos 0+0}{2+e^{0}}=\frac{1}{2+1}=\frac{1}{3}$