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Math Help - Limit of a factorial over an exponential

  1. #1
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    Limit of a factorial over an exponential

    It's been a while since I've taken Calc II, so I'm having trouble remember the exact steps to solve a problem like this:

    limit as x goes to infinity of x!/(x^x).

    I know that the bottom grows faster than the top and that the limit is nonzero, but I don't remember how to get to that answer:

    There are n factors on the top: n * (n-1) * (n-2) * ....* (n-(n-1)) * 1
    And there are n on the bottom: n * n * n .... * n

    Every factor on the top is >= n, so if you rewrite the division you get:

    (n/n) * ((n-1)/n) * ((n-2)/n) ... * (1/n)

    Half of the terms are less than 1/2 and half of the terms are greater than 1/2. Intuitively I want to say that as N goes to infinity, it's a wash and the limit is 1/2, but I don't know how to show that mathematically.

    Thanks.
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  2. #2
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    Re: Limit of a factorial over an exponential

    Quote Originally Posted by Open that Hampster! View Post
    It's been a while since I've taken Calc II, so I'm having trouble remember the exact steps to solve a problem like this:

    limit as x goes to infinity of x!/(x^x).

    I know that the bottom grows faster than the top Mr F says: Therefore it should be clear that the limit is equal to zero.

    and that the limit is nonzero, Mr F says: NO it's not. See above.

    but I don't remember how to get to that answer: Mr F says: Perhaps you should consider the gamma function ..... Furthemore, when x is an integer the analysis is not too difficult.

    There are n factors on the top: n * (n-1) * (n-2) * ....* (n-(n-1)) * 1
    And there are n on the bottom: n * n * n .... * n

    Every factor on the top is >= n, so if you rewrite the division you get:

    (n/n) * ((n-1)/n) * ((n-2)/n) ... * (1/n)

    Half of the terms are less than 1/2 and half of the terms are greater than 1/2. Intuitively I want to say that as N goes to infinity, it's a wash and the limit is 1/2, but I don't know how to show that mathematically.

    Thanks.
    ..
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  3. #3
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    Re: Limit of a factorial over an exponential

    Alright, I'm glad I haven't forgotten as much math as I thought I had.

    However, it turns out I was trying to solve a problem that didn't need to be solved. The reason the limit wasn't 0 is because I'm actually supposed to be dealing with this limit:

    lim (n->inf) lg(n!)/lg(n^n)

    So far I've gotten:

    lg(n!) - lg(n^n)
    lim lg(n!) - lim n*lg(n)

    Which is just infinity - infinity. But since n! doesn't have a derivative, L'Hospitals rule is useless, and I'm not sure I've ever learned a technique to deal with such a limit. Unless I wasn't supposed to break the fraction apart to begin with.

    EDIT: Nevermind. You can use Stirling's approximation to show that the limit is equal to 1. This thread can be closed/marked as solved/etc
    Last edited by Open that Hampster!; September 19th 2011 at 04:56 PM.
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