How to find the limit $x=\lim_{a\to{b}}{\frac{a^b-b^a}{a^a-b^b}}$ without using L'Hopital's rule? Let
$$x=\lim_{a\to{b}}{\frac{a^b-b^a}{a^a-b^b}}$$
It is very simple to solve it using L'Hopital's rule, but problem is to solve this limit without L'Hopital's rule. Is there any way to do this?

Let 
$$f(a)=a^b-b^a$$
and 
$$g(a)=a^a-b^b$$
then we have
$$x=\lim_{a\to b}\frac{f(a)-f(b)}{a-b}\times \lim_{a\to b}\left(\frac{g(a)-g(b)}{a-b}\right)^{-1}=\frac{f'(b)}{g'(b)}=\frac{1-\log b}{1+\log b}$$
