## Problem Statement

Calculate the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty} \frac{(2 n-3)^{3}-(n+5)^{3}}{(3 n-1)^{3}+(2 n+3)^{3}}
$$
## Solution

$\lim _{n \rightarrow \infty} \frac{(2 n-3)^{3}-(n+5)^{3}}{(3 n-1)^{3}+(2 n+3)^{3}}=\lim _{n \rightarrow \infty} \frac{\frac{1}{n^{3}}\left((2 n-3)^{3}-(n+5)^{3}\right)}{\frac{1}{n^{3}}\left((3 n-1)^{3}+(2 n+3)^{3}\right)}=$

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

## Problem Kuznetsov Limits 3-17