## Problem Statement

Calculate the limit of the function:

$$
\lim _{x \rightarrow 2} \sqrt[3]{\lg (x+2)+\sin \sqrt{4-x^{2}} \cdot \cos \frac{x+2}{x-2}}
$$
## Solution

Since $\cos \frac{x+2}{x-2}$ is bounded, and $\sin \sqrt{4-x^{2}} \rightarrow 0$, as $x \rightarrow 2$, then $\sin \sqrt{4-x^{2}} \cdot \cos \frac{x+2}{x-2} \rightarrow 0$, as $x \rightarrow 2$

Then:

$\lim _{x \rightarrow 2} \sqrt[3]{\lg (x+2)+\sin \sqrt{4-x^{2}} \cdot \cos \frac{x+2}{x-2}}=\sqrt[3]{\lg (2+2)+0}=\sqrt[3]{\lg 4}$