This one is from our very own archives. Just for the record you can find it here. But what kind of challenge would it be to simply apply l'Hopital's rule? So....
Find the limit without using l'Hopital's rule or power series.
-Dan
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