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Question
\[\lim_{x \to a} \frac{\cos x - \cos a}{x - a}\]
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Solution
\[ = \lim_{x \to a} \frac{- 2 \sin \left( \frac{x + a}{2} \right) \sin \left( \frac{x - a}{2} \right)}{2\left( \frac{x - a}{2} \right)} \left[ \because \cos A - \cos B - 2 \sin \left( \frac{A - B}{2} \right) \sin \left( \frac{A + B}{2} \right) \right]\]
\[ = \lim_{x \to a} - \sin \left( \frac{x + a}{2} \right) \left[ \because \lim_\theta \to a \sin\frac{\left( \theta - a \right)}{\left( \theta - a \right)} = 1 \right]\]
\[ = - \sin \left( \frac{2a}{2} \right)\]
\[ \Rightarrow - \sin a\]
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