Evaluating $\lim\limits_{x \to a} \frac{x^a - a^x}{x^a - a^a}$ without L'Hopital

Question

I'm trying to solve this limit without using L'Hopital rule. I already tried multiplying up and down by $x^a+a^a$, finding bounds for squeeze theorem, substitution of variables, but got nothing...

$a$ is a positive real number different than $1$.

Any help would be appreciated.

Answer 1

By definition of the derivative we obtain: $$\lim_{x\rightarrow a}\frac{x^a-a^x}{x^a-a^a}=\lim_{x\rightarrow a}\frac{\frac{x^a-a^a}{x-a}-\frac{a^x-a^a}{x-a}}{\frac{x^a-a^a}{x-a}}=\frac{a\cdot a^{a-1}-a^a\ln{a}}{a\cdot a^{a-1}}=1-\ln{a}.$$

Answer 2

$$=1-\dfrac{a^x-a^a}{x^a-a^a}$$

Now $$\dfrac{a^x-a^a}{x-a}=a^a\cdot\dfrac{a^{x-a}-1}{x-a}$$

Use $\lim_{h\to0}\dfrac{a^h-1}h=\ln a$ for $a>0$

Similarly for $a^{a-1}\cdot\dfrac{(x/a)^a-1}{(x/a)-1}$