Condition of the problem

Calculate the limit of the function:

$\lim _{x \rightarrow 0} \sqrt{\left(e^{\sin x}-1\right) \cos \left(\frac{1}{x}\right)+4 \cos x}$
## Solution

Since $\cos \left(\frac{1}{x}\right)_{\text {- is bounded, and }}$

$\lim _{x \rightarrow 0} e^{\sin x}-1=e^{\sin 0}-1=e^{0}-1=1-1=0$, then

$$
\left(e^{\sin x}-1\right) \cos \left(\frac{1}{x}\right) \rightarrow 0 \quad, \text { as } x \rightarrow 0
$$

Therefore:

$\lim _{x \rightarrow 0} \sqrt{\left(e^{\sin x}-1\right) \cos \left(\frac{1}{x}\right)+4 \cos x}=\sqrt{0+4 \cos 0}=\sqrt{4}=2$