## Problem Statement

Calculate the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty} \frac{(n+7)^{3}-(n+2)^{3}}{(3 n+2)^{2}+(4 n+1)^{2}}
$$
## Solution

$$
\begin{aligned}
& \lim _{n \rightarrow \infty} \frac{(n+7)^{3}-(n+2)^{3}}{(3 n+2)^{2}+(4 n+1)^{2}}=\lim _{n \rightarrow \infty} \frac{n^{3}+3 \cdot 7 \cdot n^{2}+3 \cdot 7^{2} \cdot n+7^{3}-n^{3}-3 \cdot 2 \cdot n^{2}-3 \cdot 2^{2} \cdot n-2^{3}}{(3 n+2)^{2}+(4 n+1)^{2}}= \\
& =\lim _{n \rightarrow \infty} \frac{\frac{1}{n^{2}}\left(3(7-2) n^{2}+3\left(7^{2}-2^{2}\right) n+\left(7^{3}-2^{3}\right)\right)}{\frac{1}{n^{2}}\left((3 n+2)^{2}+(4 n+1)^{2}\right)}=\lim _{n \rightarrow \infty} \frac{15+\frac{24}{n}+\frac{7^{3}-2^{3}}{n^{2}}}{\left(3+\frac{2}{n}\right)^{2}+\left(4+\frac{1}{n}\right)^{2}}= \\
& =\frac{15+0+0}{3^{2}+4^{2}}=\frac{15}{25}=\frac{3}{5}
\end{aligned}
$$

## Problem Kuznetsov Limits 3-20