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Question
\[\lim_{x \to 2} \left( \frac{1}{x - 2} - \frac{4}{x^3 - 2 x^2} \right)\]
Solution
\[\lim_{x \to 2} \left[ \frac{1}{x - 2} - \frac{4}{x^3 - 2 x^2} \right]\]
\[ = \lim_{x \to 2} \left[ \frac{1}{x - 2} - \frac{4}{x^2 \left( x - 2 \right)} \right]\]
\[ = \lim_{x \to 2} \left[ \frac{x^2 - 4}{x^2 \left( x - 2 \right)} \right]\]
\[ = \lim_{x \to 2} \left[ \frac{\left( x - 2 \right)\left( x + 2 \right)}{x^2 \left( x - 2 \right)} \right]\]
\[ = \lim_{x \to 2} \left[ \frac{x + 2}{x^2} \right]\]
\[ = \frac{2 + 2}{2^2}\]
\[ = 1\]
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