Question

Solve $\underset{x\to a}{lim}\frac{x^m-a^m}{x^n-a^n}=$

Answer 1

We have,

$\underset{x\to a}{lim}\left(\frac{x^m-a^m}{x^n-a^n}\right)$

This is $\frac{0}{0}$form.

So, apply L-Hospital rule

$\underset{x\to a}{lim}\left(\frac{m{x}^{m-1}-0}{n{x}^{n-1}-0}\right)$

$\underset{x\to a}{lim}\left(\frac{m{x}^{m-1}}{n{x}^{n-1}}\right)$

$=\frac{m{a}^{m-1}}{n{a}^{n-1}}$

Hence, the value is $\frac{m{a}^{m-1}}{n{a}^{n-1}}$.