## Problem Statement

Calculate the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty} \frac{(n+1)^{3}-(n+1)^{2}}{(n-1)^{3}-(n+1)^{3}}
$$
## Solution

$$
\begin{aligned}
& \lim _{n \rightarrow \infty} \frac{(n+1)^{3}-(n+1)^{2}}{(n-1)^{3}-(n+1)^{3}}=\lim _{n \rightarrow \infty} \frac{(n+1)^{2} \cdot((n+1)-1)}{n^{3}-3 n^{2}+3 n-1-n^{3}-3 n^{2}-3 n-1}= \\
& =\lim _{n \rightarrow \infty} \frac{n(n+1)^{2}}{-6 n^{2}-2}=\lim _{n \rightarrow \infty} \frac{n^{3}\left(1+\frac{1}{n}\right)^{2}}{-2 n^{3}\left(\frac{3}{n}+\frac{1}{n^{2}}\right)}=-\frac{1}{2} \lim _{n \rightarrow \infty} \frac{\left(1+\frac{1}{n}\right)^{2}}{\left(\frac{3}{n}+\frac{1}{n^{2}}\right)}=-\infty
\end{aligned}
$$

## Problem Kuznetsov Limits 3-6