## Problem Statement

Calculate the limit of the function:

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

Since $\sin \left(\frac{1}{x}\right)_{\text { is bounded, and }} \ln (1+x) \rightarrow 0$, as $x \rightarrow 0$, then

$$
\left(2+\sin \left(\frac{1}{x}\right)\right) \ln (1+x) \rightarrow 0 \quad \text {, as } x \rightarrow 0
$$

Then:

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