Question

The value of $\underset{x\to a}{lim}\frac{x^m-a^m}{x^n-a^n}$ equals to

• A. $\frac{m}{n}$
• B. $0$
• C. $\frac{m}{n}{a}^{m-n}$
• D. $\frac{n}{m}{a}^{n-m}$

Answer 1

The correct option is C $\frac{m}{n}{a}^{m-n}$

Using L-Hospital's rule,

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

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

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