Question

Find the following limit.
$\underset{x\to 1}{lim}\frac{x^m-1}{x^n-1}.$

Answer 1

$\underset{x\to 1}{lim}\frac{x^m-1}{x^n-1}$

here at $\underset{x\to 1}{lim}\frac{0}{0}$ form occurs.

So, applying L Hospital $\underset{x\to 1}{lim}\frac{\frac{d}{dx}(x^m-1)}{\frac{d}{dx}(x^n-1)}$

$=\underset{x\to 1}{lim}\frac{m{x}^{m-1}}{n{x}^{n-1}}$.

$=\frac{m(1{)}^{m-1}}{n(1{)}^{n-1}}$

$=\frac{m}{n}$.