\(\lim\limits_{x \to 1}\frac{x^{m-1}}{x^n-1}(\frac00 type)\)
\(=\lim\limits_{x\to 1}\frac{mx^{m-1}}{nx^{n-1}}\) (By D.L.H. Rule)
\(=\frac{m(1)^{m-1}}{n(1)^{n-1}}=\frac mn\)
Alternative:
\(\lim\limits_{x\to 1}\frac{x^{m}-1}{x^n-1}\) = \(\lim\limits_{x\to 1}\cfrac{\frac{x^m-1}{x-1}}{\frac{x^n-1}{x-1}}\)
\(=\lim\limits_{x\to 1}\frac{x^{m-1}+x^{m-2}+....x^2+x+1}{x^{n+1}+x^{n-2}+....x^2+x+1}\)
\(=\frac{1^{m+1}+1^{m-2}+1^2+1+1(m\,terms)}{1^{n-1}+1^{n-2}+...+1^2+1+1 (n\,terms)}\)
\(=\frac{1+1+...+1+1+1(m\,terms)}{1+1+...+1+1+1(n\,terms)}\)
\(=\frac mn\)