Question

Evaluate: $\underset{x\to 1}{lim}\frac{\sqrt[13]{13x}-\sqrt[7]{7x}}{\sqrt[5]{5x}-\sqrt[3]{3x}}$

Answer 1

$\underset{x\to 1}{lim}\frac{\sqrt[13]{13x}-\sqrt[7]{7x}}{\sqrt[5]{5x}-\sqrt[3]{3x}}$

Applying L.H Rule

$\underset{x\to 1}{lim}\frac{\frac{1}{13}{x}^{-\frac{12}{13}}-\frac{1}{7}{x}^{-\frac{6}{7}}}{\frac{1}{5}{x}^{-\frac{4}{5}}-\frac{1}{3}{x}^{-\frac{2}{3}}}$

$=\frac{\frac{1}{13}-\frac{1}{7}}{\frac{1}{5}-\frac{1}{3}}$

$=-\frac{6}{91}\times (-\frac{15}{2})$

$=\frac{45}{91}$