## Problem Statement

Calculate the limit of the numerical sequence:

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

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

## Problem Kuznetsov Limits 4-23