## Problem Statement

Calculate the limit of the numerical sequence:

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

$$
\begin{aligned}
& \lim _{n \rightarrow \infty} \frac{\sqrt{n+2}-\sqrt[3]{n^{3}+2}}{\sqrt[7]{n+2}-\sqrt[5]{n^{5}+2}}=\lim _{n \rightarrow \infty} \frac{\frac{1}{n}\left(\sqrt{n+2}-\sqrt[3]{n^{3}+2}\right)}{\frac{1}{n}\left(\sqrt[7]{n+2}-\sqrt[5]{n^{5}+2}\right)}= \\
& =\lim _{n \rightarrow \infty} \frac{\sqrt{\frac{1}{n}+\frac{2}{n^{2}}}-\sqrt[3]{1+\frac{2}{n^{3}}}}{\sqrt[7]{\frac{1}{n^{6}}+\frac{2}{n^{7}}}-\sqrt[5]{1+\frac{2}{n^{5}}}}=\frac{\sqrt{0+0}-\sqrt[3]{1+0}}{\sqrt[7]{0+0}-\sqrt[5]{1+0}}=\frac{-1}{-1}=1
\end{aligned}
$$

## Problem Kuznetsov Limits 4-25