If $n=p/q$, do the substitution $x=y^q$ (which is surely possible for $x>0$, with $y>0$; do the other cases separately) and $a=b^q$, so the limit becomes
$$
\lim_{y\to b}\frac{y^p-b^p}{y^q-b^q}=
\lim_{y\to b}\frac{y^p-b^p}{y-b}\frac{y-b}{y^q-b^q}=
\frac{pa^{p-1}}{qa^{q-1}}
$$
the last equality for the integer case.
I find such computations a waste of time. There's a much faster way to prove the result.
For the function $f(x)=x^{\alpha}$ (where $\alpha$ is an arbitrary real number) defined for $x>0$ we have $f(x)=\exp(\alpha\log x)$ and the chain rule says
$$
f'(x)=\exp(\alpha\log x)\frac{\alpha}{x}=x^{\alpha}\frac{\alpha}{x}=
\alpha x^{\alpha-1}
$$
For rational $\alpha$ this might be defined also for $x\le0$, but it's easy to do the case with the help of symmetries.