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Math Help - Factorial limit

  1. #1
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    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.
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  2. #2
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    Re: Factorial limit

    \frac{(x - 1)!}{x!}\ =\ \text{... ?}
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  3. #3
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    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.
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  4. #4
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    Re: Factorial limit

    Quote Originally Posted by tom@ballooncalculus View Post
    \frac{(x - 1)!}{x!}\ =\ \text{... ?}
    Haha, of course! I have no idea why I completely forgot the definition of factorial. Too much calculus...
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