## Problem Statement

Calculate the limit of the numerical sequence:

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

$$
\begin{aligned}
& \lim _{n \rightarrow \infty} \frac{n!+(n+2)!}{(n-1)!+(n+2)!}=\lim _{n \rightarrow \infty} \frac{(n-1)!(n+n(n+1)(n+2))}{(n-1)!(1+n(n+1)(n+2))}= \\
& =\lim _{n \rightarrow \infty} \frac{n+n(n+1)(n+2)}{1+n(n+1)(n+2)}=\lim _{n \rightarrow \infty} \frac{\frac{1}{n^{3}}(n+n(n+1)(n+2))}{\frac{1}{n^{3}}(1+n(n+1)(n+2))}= \\
& =\lim _{n \rightarrow \infty} \frac{\frac{1}{n^{2}}+\left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)}{\frac{1}{n^{3}}+\left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)}=\frac{0+(1+0)(1+0)}{0+(1+0)(1+0)}=1
\end{aligned}
$$

## Problem Kuznetsov Limits 6-28