Question

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

• A. $\frac{45}{91}$
• B. $\frac{90}{91}$
• C. $\frac{15}{67}$
• D. $\frac{15}{91}$

Answer 1

The correct option is A $\frac{45}{91}$

$L=\underset{x\to 1}{lim}\frac{\sqrt[13]{13x}-\sqrt[7]{7x}}{\sqrt[5]{5x}-\sqrt[3]{3x}}$
Clearly form of the limit is $\frac{0}{0}$, Thus using L-Hospital rule,
$L=\underset{x\to 1}{lim}\frac{\frac{1}{13}{x}^{1/13-1}-\frac{1}{7}{x}^{1/7-1}}{\frac{1}{5}{x}^{1/5-1}-\frac{1}{3}{x}^{1/3-1}}=\frac{\frac{1}{13}-\frac{1}{7}}{\frac{1}{5}-\frac{1}{3}}=\frac{45}{91}$