# Factorial limit

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• Jan 20th 2013, 01:05 PM
MathCrusader
Factorial limit
How do we show that

$\lim_{x \rightarrow \infty} \, \frac {e^x (x-1)!}{x!} = + \infty \, .$

I know that the factorial functionen approaches infinity faster than most other elementary functions e.g. $e^x$. But I am not sure how to think when we have a factorial function in both the nominator and denominator.
• Jan 20th 2013, 02:13 PM
tom@ballooncalculus
Re: Factorial limit
$\frac{(x - 1)!}{x!}\ =\ \text{... ?}$
• Jan 20th 2013, 02:30 PM
hollywood
Re: Factorial limit
I would start by cancelling. Since $x! = x(x-1)(x-2)\dots 1$ and $(x-1)! = (x-1)(x-2)\dots 1$, $\frac{x!}{(x-1)!}=x$ and so $\lim_{x \rightarrow \infty} \, \frac {e^x (x-1)!}{x!} = \lim_{x \rightarrow \infty} \, \frac {e^x}{x}$. And now I'll bet it looks like problems you've done before.

- Hollywood

P.S. I have to start writing faster - tom@ballooncalculus beat me to the answer.
• Jan 20th 2013, 02:34 PM
MathCrusader
Re: Factorial limit
Quote:

Originally Posted by tom@ballooncalculus
$\frac{(x - 1)!}{x!}\ =\ \text{... ?}$

Haha, of course! I have no idea why I completely forgot the definition of factorial. Too much calculus...