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Question
\[\lim_{x \to 5} \frac{x^2 - 9x + 20}{x^2 - 6x + 5}\]
Solution
\[\lim_{x \to 5} \left[ \frac{x^2 - 9x + 20}{x^2 - 6x + 5} \right]\]
\[\text{ It is of the form } \frac{0}{0} . \]
\[ \lim_{x \to 5} \left[ \frac{x^2 - 4x - 5x + 20}{x^2 - x - 5x + 5} \right]\]
\[ = \lim_{x \to 5} \left[ \frac{x\left( x - 4 \right) - 5\left( x - 4 \right)}{x\left( x - 1 \right) - 5\left( x - 1 \right)} \right]\]
\[ = \lim_{x \to 5} \left[ \frac{\left( x - 5 \right)\left( x - 4 \right)}{\left( x - 1 \right)\left( x - 5 \right)} \right]\]
\[ = \frac{5 - 4}{5 - 1}\]
\[ = \frac{1}{4}\]
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