## Problem Statement

Calculate the limit of the numerical sequence:

$\lim _{n \rightarrow \infty} \frac{(2 n+1)^{3}+(3 n+2)^{3}}{(2 n+3)^{3}-(n-7)^{3}}$
## Solution

$$
\begin{aligned}
& \lim _{n \rightarrow \infty} \frac{(2 n+1)^{3}+(3 n+2)^{3}}{(2 n+3)^{3}-(n-7)^{3}}=\lim _{n \rightarrow \infty} \frac{\frac{1}{n^{3}}\left((2 n+1)^{3}+(3 n+2)^{3}\right)}{\frac{1}{n^{3}}\left((2 n+3)^{3}-(n-7)^{3}\right)}= \\
& =\lim _{n \rightarrow \infty} \frac{\left(2+\frac{1}{n}\right)^{3}+\left(3+\frac{2}{n}\right)^{3}}{\left(2+\frac{3}{n}\right)^{3}-\left(1-\frac{7}{n}\right)^{3}}=\frac{2^{3}+3^{3}}{2^{3}-1^{3}}=\frac{35}{7}=5
\end{aligned}
$$

## Problem Kuznetsov Limits 3-19