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Question
\[\lim_{x \to 5} \frac{x^3 - 125}{x^2 - 7x + 10}\]
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Solution
\[\lim_{x \to 5} \left[ \frac{x^3 - 125}{x^2 - 7x + 10} \right]\]
\[\text{ It is of the form } \frac{0}{0} . \]
\[ \lim_{x \to 5} \left[ \frac{x^3 - 5^3}{x^2 - 2x - 5x + 10} \right]\]
\[ = \lim_{x \to 5} \left[ \frac{\left( x - 5 \right)\left( x^2 + 5x + 5^2 \right)}{x\left( x - 2 \right) - 5\left( x - 2 \right)} \right] \left[ \because A^3 - B^3 = \left( A - B \right)\left( A^2 + AB + B^2 \right) \right]\]
\[ = \lim_{x \to 5} \left[ \frac{\left( x - 5 \right)\left( x^2 + 5x + 25 \right)}{\left( x - 2 \right)\left( x - 5 \right)} \right]\]
\[ = \frac{5^2 + 5 \times 5 + 25}{5 - 2}\]
\[ = \frac{75}{3}\]
\[ = 25\]
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