Evaluate the following limit: _ x 1 1-x^ -1 3 1-x^ -2 3 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow1}\frac{1-\text{x}^{\frac{-1}{3}}}{1-\text{x}^{\frac{-2}{3}}}$

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Answer

$\lim\limits_{\text{x}\rightarrow1}\frac{1-\text{x}^{\frac{-1}{3}}}{1-\text{x}^{\frac{-2}{3}}}$
$=\lim\limits_{\text{x}\rightarrow1}\frac{1-\frac{1}{\text{x}^{\frac{1}{3}}}}{{1-\frac{1}{\text{x}^{\frac23}}}}$
$=\lim\limits_{\text{x}\rightarrow1}\frac{\Big(\text{x}^{\frac13}-1\Big)}{\Big(\text{x}^{\frac{1}{3}}-1\Big)\Big(\text{x}^{\frac13}+1\Big)}\times\text{x}^{\frac{1}{3}}$
$=\lim\limits_{\text{x}\rightarrow1}\frac{\text{x}^{\frac{1}{3}}}{\text{x}^{\frac{1}{3}}+1}$
$=\frac{1}{1+1}$
$=\frac{1}{2}$

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