## Problem Statement

Calculate the limit of the numerical sequence:

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

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

## Problem Kuznetsov Limits 4-19