## Problem Statement

Calculate the limit of the numerical sequence:

$\lim _{n \rightarrow \infty} \frac{\sqrt[4]{2+n^{5}}-\sqrt{2 n^{3}+3}}{(n+\sin n) \sqrt{7 n}}$
## Solution

$$
\begin{aligned}
& \lim _{n \rightarrow \infty} \frac{\sqrt[4]{2+n^{5}}-\sqrt{2 n^{3}+3}}{(n+\sin n) \sqrt{7 n}}=\lim _{n \rightarrow \infty} \frac{\frac{1}{n \sqrt{n}}\left(\sqrt[4]{2+n^{5}}-\sqrt{2 n^{3}+3}\right)}{\frac{1}{n \sqrt{n}}(n+\sin n) \sqrt{7 n}}= \\
& =\lim _{n \rightarrow \infty} \frac{\sqrt[4]{\frac{2}{n^{5}}+\frac{1}{n}}-\sqrt{2+\frac{3}{n^{3}}}}{\left(1+\frac{\sin n}{n}\right) \sqrt{7}}=
\end{aligned}
$$

Since $\sin n$ is bounded, then

$\frac{\sin n}{n} \rightarrow 0 \text{ as } n \rightarrow \infty$

Then:

$=\frac{\sqrt[4]{0+0}-\sqrt{2+0}}{(1+0) \sqrt{7}}=-\frac{2}{7}$