This is one of those "you have to see it before you can do it" problems. Generally if I see a limit where the exponent has a variable in it then I try out the "exponential limit":

So let's give it a try. If we set then we know that

(x is in the second quadrant, hence the negative sign.)

and thus

and the limit now goes as .

So our limit has the form:

Now we have to be a bit careful on the rigorous side of the street. Note that, before we take the limit, that the exponent is a continuous function for non-zero u. Thus we can use the following principle:

If c(u) = a(u) ^{b(u)}, then where a(u) and b(u) are continuous at the limit "point."

Thus

-Dan