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

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 , something was wrong.

But I forgot how to do it, any help?

Thank you.