## Problem Statement

Calculate the limit of the numerical sequence:

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

$$
\begin{aligned}
& \lim _{n \rightarrow \infty} \frac{\sqrt[3]{n^{2}+2}-5 n^{2}}{n-\sqrt{n^{4}-n+1}}=\lim _{n \rightarrow \infty} \frac{\frac{1}{n^{2}}\left(\sqrt[3]{n^{2}+2}-5 n^{2}\right)}{\frac{1}{n^{2}}\left(n-\sqrt{n^{4}-n+1}\right)}= \\
& =\lim _{n \rightarrow \infty} \frac{\sqrt[3]{\frac{1}{n^{4}}+\frac{2}{n^{6}}}-5}{\frac{1}{n}-\sqrt{1-\frac{1}{n^{3}}+\frac{1}{n^{4}}}}=\frac{\sqrt[3]{0+0}-5}{0-\sqrt{1-0+0}}=\frac{-5}{-1}=5
\end{aligned}
$$

## Problem Kuznetsov Limits 4-24