Answer
$\frac{m}{n}.$
Work Step by Step
We have
$$
\lim _{x \rightarrow 1} \frac{x^m-1}{x^n-1}=\frac{0}{0}
$$
This is an intermediate form, so we can apply L’Hôpital’s Rule as follows
$$
\lim _{x \rightarrow 1} \frac{x^m-1}{x^n-1}=\lim _{x \rightarrow 1} \frac{mx^{m-1}}{nx^{n-1}}=\frac{m}{n}.
$$
