## Problem Statement

Calculate the limit of the numerical sequence:

$\lim _{n \rightarrow \infty} \frac{n^{3}-(n-1)^{3}}{(n+1)^{4}-n^{4}}$
## Solution

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

## Problem Kuznetsov Limits 3-22