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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.


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Cancel out the two factorials by expanding the larger one until you get to the smaller one. In this case, $(n + 2)! = (n + 2)(n + 1)!$. – M. Vinay Jun 10 '14 at 2:52
Thanks a lot sir, that's all I needed. – MathLearner Jun 10 '14 at 3:20
up vote 7 down vote accepted

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.

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

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 $0$.

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