Question

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

Answer 1

Consider the given limit.

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

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

So, apply L-Hospital rule,

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

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

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

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

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

Hence, this is the answer.