Question

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

Answer 1

$\underset{x\to 1}{lim}\left(\frac{1-x^{-1/3}}{1-x^{-2/3}}\right)$

Let $(x^{-2/3}=y)$

$\therefore (x=y^{-3/2})$

and $x^{-1/3}=(y^{-3/2}{)}^{-1/3}$

$(x^{-1/3}=y^{1/2})$

as $x=1$

$y=(1{)}^{-2/3}$

$=1$

$\therefore \underset{y\to 1}{lim}\left(\frac{1-y^{1/2}}{1-y}\right)$

$\Rightarrow \underset{y+1}{lim}\left(\frac{1^{1/2}-y^{1/2}}{1-y}\right)$

$\Rightarrow \frac{1}{2}{y}^{1/2-1}\to \frac{1}{2}{y}^{-1/2}$

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

$\Rightarrow \frac{1}{2}(1{)}^{-1/2}\Rightarrow \frac{1}{2}(1)=\frac{1}{2}$