## Problem Statement

Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \sqrt{n(n+1)(n+2)}\left(\sqrt{n^{3}-3}-\sqrt{n^{3}-2}\right)$
## Solution

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

## Problem Kuznetsov Limits 5-30