I would want to know how we can compute the following limit by using only fundamental limits.
$$\lim_{x \to a} \dfrac{a^x-x^a}{x-a},$$ where $a$ is a positive real number.
My idea was to use a substitution: $y=x-a$. We get $$\lim\limits_{y \to 0} \dfrac{a^aa^y-(y+a)^a}{y} =a^a\left[ \lim\limits_{y \to 0} \dfrac{a^y-1+1-(\frac{y+a}{a})^a}{y} \right] =a^a\left[ \ln a+\lim\limits_{y \to 0}\frac{1-(\frac{y+a}{a})^a}{y} \right]. $$
I am looking forward to read any tips on how I can continue from this point. Any help is appreciated.