## Problem Statement

Calculate the limit of the numerical sequence:

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

$\lim _{n \rightarrow \infty} \frac{(2 n+1)^{3}-(2 n+3)^{3}}{(2 n+1)^{2}+(2 n+3)^{2}}=\lim _{n \rightarrow \infty} \frac{8 n^{3}+3 \cdot 4 n^{2}+3 \cdot 2 n+1-8 n^{3}-3 \cdot 3 \cdot 4 n^{2}-3 \cdot 3^{2} \cdot 2 n-3^{3}}{(2 n+1)^{2}+(2 n+3)^{2}}=$

$=\lim _{n \rightarrow \infty} \frac{\frac{1}{n^{2}}\left(3 \cdot 4 n^{2}(1-3)+3 \cdot 2 n\left(1-3^{2}\right)+\left(1-3^{3}\right)\right)}{\frac{1}{n^{2}}\left((2 n+1)^{2}+(2 n+3)^{2}\right)}=\lim _{n \rightarrow \infty} \frac{-24-\frac{48}{n}-\frac{26}{n^{2}}}{\left(2+\frac{1}{n}\right)^{2}+\left(2+\frac{3}{n}\right)^{2}}=$

$=\frac{-24-0-0}{2^{2}+2^{2}}=-\frac{24}{8}=-3$

## Problem Kuznetsov Limits 3-21