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Question
\[\lim_{x \to 1} \frac{1 - x^{- 1/3}}{1 - x^{- 2/3}}\]
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Solution
\[\lim_{x \to 1} \left[ \frac{1 - x^{- 1/3}}{1 - x^{- 2/3}} \right]\]
\[\text{ It is of the form } \frac{0}{0} . \]
\[ \lim_{x \to 1} \left[ \frac{\left( 1 - x^{- 1/3} \right)}{\left( 1 \right)^2 - \left( x^{- 1/3} \right)^2} \right]\]
\[ = \lim_{x \to 1} \left[ \frac{\left( 1 - x^{- 1/3} \right)}{\left( 1 - x^{- 13} \right)\left( 1 + x^{- 1/3} \right)} \right]\]
\[ = \lim_{x \to 1} \left[ \frac{1}{1 + x^{- 1/3}} \right]\]
\[ = \frac{1}{1 + 1}\]
\[ = \frac{1}{2}\]
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