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