## Problem Statement

Calculate the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty}(n \sqrt{n}-\sqrt{n(n+1)(n+2)})
$$
## Solution

$\lim _{n \rightarrow \infty}(n \sqrt{n}-\sqrt{n(n+1)(n+2)})=$

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

## Problem Kuznetsov Limits 5-26