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 $\displaystyle \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 $\displaystyle \lim _ {z \rightarrow 0 } ( \frac {1}{z^2} ) ( \frac {-1}{1-z}) = \infty (-1)$, something was wrong.

But I forgot how to do it, any help?

Thank you.