- Your last limit is incorrect. You can't just replace by . You could write instead: for any in ,
and use Taylor's theorem to show that the previous quantity converges to .
First, you wrote that , so it should be clear (from the hypotheses) that is continuous on . The only thing to check is that ( because of the previous question) .Claim: g' is continuous on U.
For any (and ), , so you have to apply Taylor's theorem to and to .