## Problem Statement

Calculate the limit of the numerical sequence:

$\lim _{n \rightarrow \infty} \frac{(n+2)^{4}-(n-2)^{4}}{(n+5)^{2}+(n-5)^{2}}$
## Solution

$$
\begin{aligned}
& \lim _{n \rightarrow \infty} \frac{(n+2)^{4}-(n-2)^{4}}{(n+5)^{2}+(n-5)^{2}}=\lim _{n \rightarrow \infty} \frac{\left((n+2)^{2}-(n-2)^{2}\right) \cdot\left((n+2)^{2}+(n-2)^{2}\right)}{(n+5)^{2}+(n-5)^{2}}= \\
& =\lim _{n \rightarrow \infty} \frac{\left(n^{2}+4 n+4-n^{2}+4 n-4\right)\left(n^{2}+4 n+4+n^{2}-4 n+4\right)}{(n+5)^{2}+(n-5)^{2}}= \\
& \lim _{n \rightarrow \infty} \frac{\frac{1}{n^{2}} 8 n\left(2 n^{2}+8\right)}{\frac{1}{n^{2}}\left((n+5)^{2}+(n-5)^{2}\right)}=\lim _{n \rightarrow \infty} \frac{16 n\left(1+\frac{4}{n^{2}}\right)}{\left(1+\frac{5}{n}\right)^{2}+\left(1-\frac{5}{n}\right)^{2}}=+\infty
\end{aligned}
$$

## Problem Kuznetsov Limits 3-23