
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 $\displaystyle \lim _ {z \rightarrow 0 } (1z)^{ \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 $\displaystyle \lim _ {z \rightarrow 0 } ( \frac {1}{z^2} ) ( \frac {1}{1z}) = \infty (1)$, something was wrong.
But I forgot how to do it, any help?
Thank you.

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