Thank you ksdhart and Ishuda!
ksdhart - Yeah, it looks like I will have to do some l'hopitaling. I hadn't considered that in the angle addition form so that was a great idea.
Ishuda - I think that I will have to take the other terms of the fraction into consideration because the final answer should be some non-zero, non-infinite function of theta. There is a chance that the steps that lead to taking this limit have a error, but I have checked several times, so I don't think so. (there is a link to my original post that describes the larger problem that I am working on.)
I'll post back here with any more progress (hopefully this weekend.)
Thanks again,
Jake