I came up with a nice little limit for the Challenge Problems section. Unfortunately I can't actually solve the thing, so it's not a candidate. But it's been driving me crazy. Here it is:
and, of course, no l'Hopital's or series expansions. I even tried Wolfram|Alpha and it did it by l'Hopital's.
I've got two approaches which might bear fruit but I don't quite know how to apply them. The first is using the squeeze theorem, and the second is a good old fashioned epsilon-delta method. The problem here is that the function is discontinuous at the origin and I can't see how to apply either method.