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?

Well, this is essentally the long form of L'Hopital, but:
$$\begin{align}\frac{a^b-b^a}{a^a-b^b} &= \frac{a^b-b^b+b^b-b^a}{a-b} \cdot\frac{a-b}{a^a-b^b}\\
&=\frac{\dfrac{a^b-b^b}{a-b} - \dfrac{b^a-b^b}{a-b}}{\dfrac{a^a-b^b}{a-b}}
\end{align}$$
The three terms here converges to the derivative of the functions $x^b$, $b^x$, and $x^x$ at $x=b$.
