meromorphic map and poles

1) Let f be meromorphic in C. Show that if f(1/z) is also meromorphic then f is the quotient of two polynomials.

2) Let a be an isolated singularity of f. Then exp(f(z)) has no pole in a.

Regarding 1), I am quite clueless. Regarding 2) it is clear that if a is avoidable for f, then it is avoidable for exp o f, because exp o g is an holomorphic extension of exp o f, if g is one of f.

But if a is a pole of f, I cant see why it should not be one of exp o f, for isnt lim_z->a e^f(z) = oo, because lim_z->a f(z) is oo?. I guess I shoul try and prove it is an esential singularity, as happens with e^(1\z) in 0.

Thanks