Question

$\underset{x\to a}{\lim }\frac{x^{2/3}-a^{2/3}}{x^{3/4}-a^{3/4}}$

Answer 1

$$\begin{gathered} \underset{x\to a}{\lim }\left[\frac{x^{{\displaystyle \frac{2}{3}}}-a^{{\displaystyle \frac{2}{3}}}}{x^{{\displaystyle \frac{3}{4}}}-a^{{\displaystyle \frac{3}{4}}}}\right] \\ =\underset{a\to a}{\lim }\left[\left(\frac{x^{{\displaystyle \frac{2}{3}}}-a^{{\displaystyle \frac{2}{3}}}}{x-a}\right)\times \left(\frac{x-a}{x^{{\displaystyle \frac{3}{4}}}-a^{{\displaystyle \frac{3}{4}}}}\right)\right] \\ =\frac{2}{3}{\left(a\right)}^{\frac{2}{3}-1}\times \frac{1}{{\displaystyle \frac{3}{4}}{a}^{{\displaystyle \frac{3}{4}}-1}} \\ =\frac{8}{9}\frac{a^{-{\displaystyle \frac{1}{3}}}}{a^{-{\displaystyle \frac{1}{4}}}} \\ =\frac{8}{9}{a}^{-\frac{1}{3}+\frac{1}{4}} \\ =\frac{8}{9}{a}^{-\frac{1}{12}} \end{gathered}$$