## Problem Statement

Calculate the limit of the numerical sequence:

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

$$
\begin{aligned}
& \lim _{n \rightarrow \infty} \sqrt{n+2}(\sqrt{n+3}-\sqrt{n-4})= \\
& =\lim _{n \rightarrow \infty} \frac{\sqrt{n+2}(\sqrt{n+3}-\sqrt{n-4})(\sqrt{n+3}+\sqrt{n-4})}{\sqrt{n+3}+\sqrt{n-4}}= \\
& =\lim _{n \rightarrow \infty} \frac{\sqrt{n+2}(n+3-(n-4))}{\sqrt{n+3}+\sqrt{n-4}}=\lim _{n \rightarrow \infty} \frac{7 \sqrt{n+2}}{\sqrt{n+3}+\sqrt{n-4}}=
\end{aligned}
$$

$$
\begin{aligned}
& =\lim _{n \rightarrow \infty} \frac{\frac{1}{\sqrt{n}} 7 \sqrt{n+2}}{\frac{1}{\sqrt{n}}(\sqrt{n+3}+\sqrt{n-4})}=\lim _{n \rightarrow \infty} \frac{7 \sqrt{1+\frac{2}{n}}}{\sqrt{1+\frac{3}{n}}+\sqrt{1-\frac{4}{n}}}= \\
& =\frac{7 \sqrt{1+0}}{\sqrt{1+0}+\sqrt{1-0}}=\frac{7}{2}
\end{aligned}
$$

## Problem Kuznetsov Limits 5-28