L'Hospital Rule with exponentials help.

[xlxJesus]

Hello, so I'm currently in Calculus and am going over test corrections for extra credit. I'm stuck on a problem that requires L'Hospital Rule. I've tried taking the derivative but from there I just seem to get stuck at a dead end. It's multiple choice so it's easier to narrow down the correct answer.
The problem should be attached below.

 

[Dr.Peterson]

Please show complete work, including the answer you get.

I hope you realize that L'Hopital does not apply directly to this problem, so you have to do something else first. What is it? And you don't just take the derivative; you take two derivatives. What are they?