It makes sense to me that the limit doesn't exist since (1/x) becomes arbitrarily large as x->0 but I'm a little unsure how to go about proving this. I can start by breaking it up into two separate limits, but I'm not sure if that will help at all. That will just give me 0*infinity which is indeterminate.

I think I'll split it and look at tan(1/x) as sin(1/x)/cos(1/x).