## Problem Statement

Calculate the limit of the numerical sequence:

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

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

## Problem Kuznetsov Limits 4-3