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Question
\[\lim_{x \to 27} \frac{\left( x^{1/3} + 3 \right) \left( x^{1/3} - 3 \right)}{x - 27}\]
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Solution
\[\lim_{x \to 27} \frac{\left[ x^\frac{1}{3} + 3 \right] \left[ x^\frac{1}{3} - 3 \right]}{x - 27}\]
\[ = \lim_{x \to 27} \left[ \frac{\left( x^\frac{1}{3} + 3 \right) \left( x^\frac{1}{3} - 3 \right)}{\left( x^\frac{1}{3} \right)^3 - 3^3} \right]\]
\[ x \to 27\]
\[ \therefore x^\frac{1}{3} \to 3\]
\[Let y = x^\frac{1}{3} \]
\[ \lim_{y \to 3} \left[ \frac{\left( y + 3 \right) \left( y - 3 \right)}{y^3 - 3^3} \right]\]
\[ = \frac{\left( 3 + 3 \right)}{3 \times 3^{3 - 1}}\]
\[ = \frac{6}{3 \times 9}\]
\[ = \frac{2}{9}\]
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