## Problem Statement

Calculate the limit of the numerical sequence:

$\lim _{n \rightarrow \infty} \frac{n \sqrt[3]{5 n^{2}}+\sqrt[4]{9 n^{8}+1}}{(n+\sqrt{n}) \sqrt{7-n+n^{2}}}$
## Solution

$$
\begin{aligned}
& \lim _{n \rightarrow \infty} \frac{n \sqrt[3]{5 n^{2}}+\sqrt[4]{9 n^{8}+1}}{(n+\sqrt{n}) \sqrt{7-n+n^{2}}}=\lim _{n \rightarrow \infty} \frac{\frac{1}{n^{2}}\left(n \sqrt[3]{5 n^{2}}+\sqrt[4]{9 n^{8}+1}\right)}{\frac{1}{n^{2}}(n+\sqrt{n}) \sqrt{7-n+n^{2}}}= \\
& =\lim _{n \rightarrow \infty} \frac{\sqrt[3]{\frac{5}{n}}+\sqrt[4]{9+\frac{1}{n^{8}}}}{\left(1+\sqrt{\frac{1}{n}}\right) \sqrt{\frac{7}{n^{2}}-\frac{1}{n}+1}}=\frac{\sqrt[3]{0}+\sqrt[4]{9+0}}{(1+\sqrt{0}) \sqrt{0-0+1}}=\frac{\sqrt{3}}{1}=\sqrt{3}
\end{aligned}
$$

## Problem Kuznetsov Limits 4-1