<s>[INST] 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?
 [/INST] 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 $x=b$.
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