## Problem Statement

Calculate the limit of the numerical sequence:

$\lim _{n \rightarrow \infty} \frac{n^{2}-\sqrt{3 n^{5}-7}}{\left(n^{2}-n \cos n+1\right) \sqrt{n}}$
## Solution

$\lim _{n \rightarrow x} \frac{n^{2}-\sqrt{3 n n^{5}-\overline{7}}}{\left(n^{2}-n \cos n+1\right) \sqrt{n}}=\lim _{n \rightarrow x} \frac{\frac{1}{n^{2} \sqrt{n}}\left(n^{2}-\sqrt{3 n l^{5}-\bar{i}}\right)}{\frac{1}{n^{2} \sqrt{n}}\left(n^{2}-n \cos n+1\right) \sqrt{n}}=$

$=\lim _{n \rightarrow x} \frac{\frac{1}{\sqrt{n}}-\sqrt{3-\frac{7}{n^{3}}}}{1-\frac{\cos n}{n}+\frac{1}{n^{2}}}=$

Since $\cos n$ is bounded, then

$$
\frac{\cos n}{n} \rightarrow 0, \text { as } n \rightarrow \infty
$$

Then

$$
=\frac{0-\sqrt{3-0}}{1-0+0}=-\sqrt{3}
$$