Limits with factorial

Question

I'm having difficulties understanding all limits with factorial...

Actually, what I don't understand is not the limit concept but how to simplify factorial...

Example :

$$\lim\limits_{n \to \infty} \frac{(n+1)!((n+1)^2 + 1)}{(n^2+1)(n+2)!}$$

I know that it's supposed to give $0$ as I have the answer, but I'd like to understand how to do it as each time I get a limit with factorial I get stuck.

Thanks.

Answer 1

You have $\frac{(n+1)!}{(n+2)!}$ as the only two factorials in the limit. This can be re-written as $\frac{(n+1)!}{(n+2)(n+1)!}$ or $\frac{1}{(n+2)}$. After you simplify this section, the rest of the limit should be relatively easy to calculate, given a basic understanding of limits.

Answer 2

$\frac{(n+1)!}{(n+2)!} = \frac{1}{n+2}$

So, the problem reduces to $lim_{n\rightarrow \infty} \frac{(n+1)^2 + 1)}{(n+2)(n^2 + 1)}$.

The numerator is quadratic in $n$ while the denominator is cubic, so as $n \rightarrow \infty$ the limit should go to $0$.