How to find $\lim_{x \to a}\frac{ a^nf(x)-x^nf(a)}{x-a}$

The Quotient Rule idea is good, and is carried out very well. The only problem is that it does not work in the case $a=0$. But that case is easily dealt with separately.

A simpler approach is to note that $$a^nf(x)-x^nf(a)=(a^n f(x)-a^nf(a))-(x^nf(a)-a^nf(a)).$$

Another possible way could be Taylor expansion around $x=a$ $$x^n=a^n+n a^{n-1} (x-a)+O\left((x-a)^2\right)$$ $$f(x)=f(a)+(x-a) f'(a)+O\left((x-a)^2\right)$$ Then $$a^n f(x)-x^nf(a)=(x-a) \left(a^n f'(a)-n a^{n-1}f(a)\right)+O\left((x-a)^2\right)$$