Two facts of limit

**tttcomrader** · August 9th 2009, 08:15 PM

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'Hôpital'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.

**Random Variable** · August 9th 2009, 08:37 PM

$\lim_{z \to 0} \ln (1-z)^{-1/z}$

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

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

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

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

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

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