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Math Help - Two facts of limit

  1. #1
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    A simple limit fact

    I'm reading some probability and statistics, and ran into two limit facts when I get to the part of Poisson Probability Function proof.

    It says that  \lim _ {z \rightarrow 0 } (1-z)^{- \frac {1}{z} } = e

    It is almost embarrassing for me to ask, as I do remember encountering this problem when I took calc, and you would expect someone who finish Real Analysis would be able to solve them.

    So far, for the first one, I used the l'H˘pital's rule with natural log, but then I have  \lim _ {z \rightarrow 0 } ( \frac {1}{z^2} ) ( \frac {-1}{1-z}) = \infty (-1), something was wrong.



    But I forgot how to do it, any help?

    Thank you.
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  2. #2
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     \lim_{z \to 0} \ln (1-z)^{-1/z}

     =\lim_{z \to 0} \frac{-1}{z} \ln (1-z)

     = \lim_{z \to 0} \frac{\ln (1-z)}{-z}

     = \lim_{z \to 0} \frac{\frac{1}{1-z}(-1)}{-1}

     = \lim_{z \to 0} \frac {1}{1-z} = 1

    so  \lim_{z \to 0} (1-z)^{-1/z} = e^{1} = e
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