Question

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

Answer 1

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

$={lim}_{x\to a}\frac{\frac{x^{\frac{5}{7}}-a^{\frac{5}{7}}}{x-a}}{\frac{x^{\frac{2}{7}}-a^{\frac{2}{7}}}{x-a}}$

[Dividing numerator and denominator by x-a]

$=\frac{{lim}_{x\to a}\frac{x^{\frac{5}{7}-a^{\frac{5}{7}}}}{x-a}}{{lim}_{x\to a}\frac{x^{\frac{2}{7},-a^{\frac{2}{7}}}}{x-a}}$

Applying formula ${lim}_{x\to a}\frac{x^n-a^n}{x-a}=n{a}^{n-1}$

$=\frac{\frac{5}{7}(a{)}^{\frac{5}{7}-1}}{\frac{2}{7}(a{)}^{\frac{2}{7}-1}}$

$=\frac{\frac{5}{7}{a}^{\frac{-2}{7}}}{\frac{2}{7}{a}^{\frac{-5}{7}}}$

$=\frac{5}{7}{a}^{\frac{-2}{7}+\frac{5}{7}}$

$=\frac{5}{7}{a}^{\frac{3}{7}}$