## Problem Statement

Calculate the limit of the numerical sequence:

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

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

## Problem Kuznetsov Limits 4-30