Quote by Avodyne
This is not helpful because x is not always close to 0 as you integrate. Can you see a change variables that would help?

I don't see how a change of variable would help. The OP now has to argue that the contributions to the integral in the limit from outside of [delta,delta] can also be bounded by epsilon by taking alpha sufficiently large. Exactly how you would technically do this depends on what additional conditions you want to put on g(x). And you do need something. If g(x)=exp(x^4) then the limit doesn't even exist.