## Problem Statement

Calculate the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty} n\left(\sqrt{n^{4}+3}-\sqrt{n^{4}-2}\right)
$$
## Solution

$\lim _{n \rightarrow \infty} n\left(\sqrt{n^{4}+3}-\sqrt{n^{4}-2}\right)=$

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

## Problem Kuznetsov Limits 5-29