limit involving exponential

Hello,

I've lurked this forum before but this is my first time posting. I came across a limit this morning that I feel like I really SHOULD be able to answer but can't. I think I'm missing something stupid. Especially since I think this particular limit comes up a bunch.

I've found out that it equals e (from online calculators (Headbang)) but I have no idea why. I've tried converting it into a rational and using l'Hopital's rule, but I'm getting stuck. Am I on the right track? Any hints?