## Problem Statement

Calculate the limit of the numerical sequence:

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

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

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

$$
\begin{aligned}
& =\lim _{n \rightarrow \infty}\left(\frac{(2 n+1)!}{(2 n+2)!\cdot(2 n+2)}+\frac{(2 n+2)!}{(2 n+2)!\cdot(2 n+2)}\right)= \\
& =\lim _{n \rightarrow \infty}\left(\frac{1}{(2 n+2) \cdot(2 n+2)}+\frac{1}{2 n+2}\right)=0+0=0
\end{aligned}
$$