Question

$\underset{x\to a}{lim}\frac{x^m-a^m}{x^n-a^n}$ is equal to.

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

Answer 1

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

We have, $\underset{x\to a}{lim}\frac{x^m-a^m}{x^n-a^n}$

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

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

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

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