Question

Solve:
$displaystyle{lim}_{x\to 0}\frac{(a+x{)}^{2/3}-a^{2/3}}{x}$

Answer 1

Consider the following question.

${lim}_{x\to 0}\frac{{(a+x)}^{\frac{2}{3}}-a^{\frac{2}{3}}}{x}$

$={lim}_{x\to 0}\frac{2}{3}\frac{{(a+x)}^{\frac{-1}{3}}}{1}$

$=\frac{2}{3}\frac{{(a+0)}^{\frac{-1}{3}}}{1}$

$=\frac{2}{3}\frac{1}{a^{\frac{1}{3}}}$

Hence, this is the required answer.