## Problem Statement

Calculate the limit of the numerical sequence:

$\lim _{n \rightarrow \infty} \frac{\sqrt{n+6}-\sqrt{n^{2}-5}}{\sqrt[3]{n^{3}+3}+\sqrt[4]{n^{3}+1}}$
## Solution

$$
\begin{aligned}
& \lim _{n \rightarrow \infty} \frac{\sqrt{n+6}-\sqrt{n^{2}-5}}{\sqrt[3]{n^{3}+3}+\sqrt[4]{n^{3}+1}}=\lim _{n \rightarrow \infty} \frac{\frac{1}{n}\left(\sqrt{n+6}-\sqrt{n^{2}-5}\right)}{\frac{1}{n}\left(\sqrt[3]{n^{3}+3}+\sqrt[4]{n^{3}+1}\right)}= \\
& =\lim _{n \rightarrow \infty} \frac{\sqrt{\frac{1}{n}+\frac{6}{n^{2}}}-\sqrt{1-\frac{5}{n^{2}}}}{\sqrt[3]{1+\frac{3}{n^{3}}}+\sqrt[4]{\frac{1}{n}+\frac{1}{n^{4}}}}=\frac{\sqrt{0+0}-\sqrt{1-0}}{\sqrt[3]{1+0}+\sqrt[4]{0+0}}=\frac{-1}{1}=-1
\end{aligned}
$$

## Problem Kuznetsov Limits 4-27