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Math Help - Exponential Limit

  1. #1
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    Exponential Limit

    Hi all,

    I'm trying to evaluate the limit of (a(1-exp(a)))/(n(1-exp(a/n))) as n goes to infinity and 'a' is constant.

    I think that you have to rearrange and then use l'hoptials rule somehow, but i'm unsure. Any help would be greatly appreciated.

    Thanks.
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  2. #2
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    Write it as \frac{f(x)}{g(x)} where f(x)={\frac{a(1-e^a)}{n}} and g(x)=e^{1- \displaystyle\frac{a}{n}}.
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  3. #3
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by DeFacto View Post
    Hi all,

    I'm trying to evaluate the limit of (a(1-exp(a)))/(n(1-exp(a/n))) as n goes to infinity and 'a' is constant.

    I think that you have to rearrange and then use l'hoptials rule somehow, but i'm unsure. Any help would be greatly appreciated.

    Thanks.
    \frac{a(1-e^a)}{n(1-e^{a/n})} = \frac{\frac{a(1-e^a)}{n}}{1-e^{a/n}} = \frac{f(n)}{g(n)}

    Use L'Hopital's Rule:

    Spoiler:
    \frac{f'(n)}{g'(n)} = \frac{-\frac{a(1-e^a)}{n^2}}{-e^{a/n}\cdot -a/n^2} = -\frac{a(1-e^a)}{ae^{a/n}}

    \lim_{n\to\infty}-\frac{a(1-e^a)}{ae^{a/n}} = -\frac{a(1-e^a)}{a} = \boxed{e^a-1}
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