Hello everyone I was wondering if anyone could give me a hand with this

Let f be an entire function such that as | . Define g by

where

Show that g has a pole at 0, and, by considering the Laurent series for g about 0, show that f is a polynomial function.

Im not so worried about the 2nd part yet but i was wondering how to show that g has a pole, intuitvely i can see that it has one at zero

i am trrying to express g as

where h(z) is analytic on some open disc around 0 and

but im not sure how to do this

any help would be appreciated, thank you :)