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Question
\[\lim_{x \to a} \frac{x^{5/7} - a^{5/7}}{x^{2/7} - a^{2/7}}\]
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Solution
\[\lim_{x \to a} \left[ \frac{x^\frac{5}{7} - a^\frac{5}{7}}{x^\frac{2}{7} - a^\frac{2}{7}} \right]\]
\[ = \lim_{x \to a} \left[ \left( \frac{x^\frac{5}{7} - a^\frac{5}{7}}{x - a} \right) \times \left( \frac{x - a}{x^\frac{2}{7} - a^\frac{2}{7}} \right) \right]\]
\[ = \frac{5}{7} a^\frac{5}{7} - 1 \times \frac{1}{\frac{2}{7} a^\frac{2}{7} - 1}\]
\[ = \frac{5}{2} a^{- \frac{2}{7}} a^\frac{5}{7} \]
\[ = \frac{5}{2} a^{- \frac{2}{7} + \frac{5}{7}} \]
\[ = \frac{5}{2} a^\frac{3}{7}\]
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