Answer
$\dfrac{a}{b}$
Work Step by Step
Consider $f(x)=\lim\limits_{x \to 1}\dfrac{x^a-1}{x^b-1}=\lim\limits_{x \to 1} \dfrac{a(x)}{b(x)}$ and $a(1)=0, b(1)=0$
Thus, $f(1)=\dfrac{0}{0}$
This shows an Inderminate form of the limit, so apply L-Hospital's rule:
$\lim\limits_{x \to l}\dfrac{a(x)}{b(x)}=\lim\limits_{x \to l}\dfrac{a'(x)}{b'(x)}$
Thus,
$\lim\limits_{x \to 1}\dfrac{ax^{a-1}}{bx^{b-1}}=\dfrac{a}{b}$
