Condition of the problem

Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{e^{1 / n}+\sin \left(\frac{n}{n^{2}+1}\right) \cdot \cos n}{1+\cos \left(\frac{1}{n}\right)}$
## Solution

Since $\cos n$ is bounded, and $\sin \left(\frac{n}{n^{2}+1}\right) \rightarrow 0$ as $n \rightarrow \infty$, then

$$
\sin \left(\frac{n}{n^{2}+1}\right) \cdot \cos n \rightarrow 0, \text { as } n \rightarrow \infty
$$

Then:

$$
\lim _{n \rightarrow \infty} \frac{e^{1 / n}+\sin \left(\frac{n}{n^{2}+1}\right) \cdot \cos n}{1+\cos \left(\frac{1}{n}\right)}=\frac{e^{0}+0}{1+\cos (0)}=\frac{1}{1+1}=\frac{1}{2}
$$