Question

Evaluate : $\underset{x\to 1}{lim}\frac{1-x^{-1/3}}{1-x^{-2/3}}$

Answer 1

Consider the following question.

${lim}_{x\to 1}\frac{1-x^{\frac{-1}{3}}}{1-x^{\frac{-2}{3}}}\frac{}{}$

Apply L-Hospital rule,

$={lim}_{x\to 1}\frac{\frac{1}{3}{x}^{\frac{-4}{3}}}{\frac{2}{3}{x}^{{}^{\frac{-5}{3}}}}$

$={lim}_{x\to 1}\frac{\frac{1}{3}{\left(1\right)}^{\frac{-4}{3}}}{\frac{2}{3}{\left(1\right)}^{{}^{\frac{-5}{3}}}}$

$=2$

Hence, this is the required answer.