## Problem Statement

Calculate the limit of the numerical sequence:

$\lim _{n \rightarrow \infty} \frac{\sqrt{n^{8}+6}-\sqrt{n-6}}{\sqrt[8]{n^{8}+6}+\sqrt{n-6}}$
## Solution

$$
\begin{aligned}
& \lim _{n \rightarrow \infty} \frac{\sqrt{n^{8}+6}-\sqrt{n-6}}{\sqrt[8]{n^{8}+6}+\sqrt{n-6}}=\lim _{n \rightarrow \infty} \frac{\frac{1}{n}\left(\sqrt{n^{8}+6}-\sqrt{n-6}\right)}{\frac{1}{n}\left(\sqrt[8]{n^{8}+6}+\sqrt{n-6}\right)}= \\
& =\lim _{n \rightarrow \infty} \frac{\sqrt{n^{6}+\frac{6}{n^{2}}}-\sqrt{\frac{1}{n}-\frac{6}{n^{2}}}}{\sqrt[8]{1+\frac{6}{n^{8}}}+\sqrt{\frac{1}{n}-\frac{6}{n^{2}}}}=\left\{\frac{\sqrt{\infty+0}-\sqrt{0-0}}{\sqrt[8]{1+0}+\sqrt{0-0}}=\frac{\infty}{1}\right\}=\infty
\end{aligned}
$$

## Problem Kuznetsov Limits 4-28