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Question
\[\lim_{x \to a} \frac{x^{2/3} - a^{2/3}}{x^{3/4} - a^{3/4}}\]
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Solution
\[\lim_{x \to a} \left[ \frac{x^\frac{2}{3} - a^\frac{2}{3}}{x^\frac{3}{4} - a^\frac{3}{4}} \right]\]
\[ = \lim_{a \to a} \left[ \left( \frac{x^\frac{2}{3} - a^\frac{2}{3}}{x - a} \right) \times \left( \frac{x - a}{x^\frac{3}{4} - a^\frac{3}{4}} \right) \right]\]
\[ = \frac{2}{3} \left( a \right)^\frac{2}{3} - 1 \times \frac{1}{\frac{3}{4} a^\frac{3}{4} - 1}\]
\[ = \frac{8}{9} \frac{a^{- \frac{1}{3}}}{a^{- \frac{1}{4}}}\]
\[ = \frac{8}{9} a^{- \frac{1}{3} + \frac{1}{4}} \]
\[ = \frac{8}{9} a^{- \frac{1}{12}}\]
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