Question

$\underset{x\to a}{\lim }\frac{x^{5/7}-a^{5/7}}{x^{2/7}-a^{2/7}}$

Answer 1

$$\begin{gathered} \underset{x\to a}{\lim }\left[\frac{x^{{\displaystyle \frac{5}{7}}}-a^{{\displaystyle \frac{5}{7}}}}{x^{{\displaystyle \frac{2}{7}}}-a^{{\displaystyle \frac{2}{7}}}}\right] \\ =\underset{x\to a}{\lim }\left[\left(\frac{x^{{\displaystyle \frac{5}{7}}}-a^{{\displaystyle \frac{5}{7}}}}{x-a}\right)\times \left(\frac{x-a}{x^{{\displaystyle \frac{2}{7}}}-a^{{\displaystyle \frac{2}{7}}}}\right)\right] \\ =\frac{5}{7}{a}^{\frac{5}{7}-1}\times \frac{1}{{\displaystyle \frac{2}{7}}{a}^{{\displaystyle \frac{2}{7}}-1}} \\ =\frac{5}{2}{a}^{-\frac{2}{7}}{a}^{\frac{5}{7}} \\ =\frac{5}{2}{a}^{-\frac{2}{7}+\frac{5}{7}} \\ =\frac{5}{2}{a}^{\frac{3}{7}} \end{gathered}$$