I have this really nasty limit that I've been banging my head on all night. The question asks me to calculate the following:

The limit evaluates to , so my first instinct was to use L'Hospital's Rule and take the limit of . But that didn't really simplify it; doing that just gave another nasty limit. Is there a simpler approach?

According to Wolfram Alpha, the limit is equal to . But it just says "series expansion at " without showing any of the steps.

I know that is the sum of an infinite series. But how do you recognize that from the limit? And how do you get to ?

Any pointers would be greatly appreciated!

- A really puzzled math student.