Question

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

Answer 1

$$\begin{gathered} \underset{x\to 1}{\lim }\left[\frac{1-x^{-1/3}}{1-x^{-2/3}}\right] \\ \mathrm{It}\mathrm{is}\mathrm{of}\mathrm{the}\mathrm{form}\frac{0}{0}. \\ \underset{x\to 1}{\lim }\left[\frac{\left(1-x^{-1/3}\right)}{{\left(1\right)}^2-{\left(x^{-1/3}\right)}^2}\right] \\ =\underset{x\to 1}{\lim }\left[\frac{\left(1-x^{-1/3}\right)}{\left(1-x^{-13}\right)\left(1+x^{-1/3}\right)}\right] \\ =\underset{x\to 1}{\lim }\left[\frac{1}{1+x^{-1/3}}\right] \\ =\frac{1}{1+1} \\ =\frac{1}{2} \end{gathered}$$