Answer
$na^{n-1}$
Work Step by Step
We have to determine $L=\lim\limits_{x \to a} \dfrac{x^n-a^n}{x-a}$, where $n$ is a positive integer.
Use the factorization formula:
$x^n-a^n=(x-a)(x^{n-1}+x^{n-2}a+......+xa^{n-2}+a^{n-1})$
$L=\lim\limits_{x \to a} \dfrac{(x-a)(x^{n-1}+x^{n-2}a+......+xa^{n-2}+a^{n-1})}{x-a}$
Simplify:
$L=\lim\limits_{x \to a} (x^{n-1}+x^{n-2}a+......+xa^{n-2}+a^{n-1})$
Determine the limit:
$L=a^{n-1}+a^{n-2}(a)+a^{n-2}(a^2)+...+a(a^{n-2})+a^{n-1}=a^{n-1}+a^{n-1}+a^{n-1}+...+a^{n-1}=na^{n-1}$
