## Problem Statement

Calculate the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty} \frac{\sqrt{n+3}-\sqrt[3]{8 n^{3}+3}}{\sqrt[4]{n+4}-\sqrt[5]{n^{5}+5}}
$$
## Solution

$$
\begin{aligned}
& \lim _{n \rightarrow \infty} \frac{\sqrt{n+3}-\sqrt[3]{8 n^{3}+3}}{\sqrt[4]{n+4}-\sqrt[5]{n^{5}+5}}=\lim _{n \rightarrow \infty} \frac{\frac{1}{n}\left(\sqrt{n+3}-\sqrt[3]{8 n^{3}+3}\right)}{\frac{1}{n}\left(\sqrt[4]{n+4}-\sqrt[5]{n^{5}+5}\right)}= \\
& =\lim _{n \rightarrow \infty} \frac{\sqrt{\frac{1}{n}+\frac{3}{n^{2}}}-\sqrt[3]{8+\frac{3}{n^{3}}}}{\sqrt[4]{\frac{1}{n^{3}}+\frac{4}{n^{4}}}-\sqrt[5]{1+\frac{5}{n^{5}}}}=\frac{\sqrt{0+0}-\sqrt[3]{8+0}}{\sqrt[4]{0+0}-\sqrt[5]{1+0}}=\frac{-2}{-1}=2
\end{aligned}
$$

## Problem Kuznetsov Limits 4-20