Question

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

Answer 1

$\underset{x\to a}{lim}\frac{x^m-a^m}{x^n-a^n}$ is of the indeterminate form $\frac{0}{0}$

So, we can apply L'Hospital rule.

$\frac{d}{dx}(x^m-a^m)=m{x}^{m-1}-0$

$=m{x}^{-m-1}$

$\frac{d}{dx}(x^n-a^n)=n{x}^{n-1}-0=n{x}^{n-1}$

$\underset{x\to a}{lim}\frac{x^m-a^m}{x^n-a^n}=\underset{x\to a}{lim}\frac{\frac{d}{dx}(x^m-a^m)}{\frac{d}{dx}(x^n-a^n)}$

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