TITLE: How to find the limit of function $\lim_{x\to 0} \frac {a^b-b^a} {a-b}$
QUESTION [5 upvotes]: Let $a(x)$ and $b(x)$ be continuous functions such that $a(0)=b(0)=3$  
How to find the limit: $$\lim_{x\to 0} \frac {a^b-b^a} {a-b}$$  
I tried to annihilate $b-a$ by writing $a^b-b^a= \sum_{k=0}^{\infty} \frac {{(b\ln a)}^k-({a\ln b)}^k} {k!}$ but it leads to another limit $\lim \frac {b\ln a-a\ln b} {a-b}$ that seems to be similar to the first one...

REPLY [2 votes]: In the general case, if $\lim_{x\to0}a(x) = \lim_{x\to0}b(x) = y$, then for any function $f$ differentiable at $y$ you have
$$\lim_{x\to0} \frac{f(a(x))-f(b(x))}{a(x)-b(x)} = f'(y).$$
Now, on your case you have
$$\frac{a^b-b^a}{a-b}=-\frac{a^a-a^b}{a-b}+\frac{a^a-b^b}{a-b}-\frac{b^a-b^b}{a-b}.$$
I think you can conclude from there on ;)