## Problem Statement

Calculate the limit of the numerical sequence:

$\lim _{n \rightarrow \infty} \frac{n^{2}-\sqrt{n^{3}+1}}{\sqrt[3]{n^{6}+2}-n}$
## Solution

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

## Problem Kuznetsov Limits 4-29