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Question
\[\lim_{x \to a} \frac{\cos x - \cos a}{\sqrt{x} - \sqrt{a}}\]
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Solution
\[\lim_{x \to a} \frac{\cos x - \cos a}{\sqrt{x} - \sqrt{a}}\]
\[ = \lim_{x \to a} \frac{- 2 \sin \left( \frac{x + a}{2} \right) \sin \left( \frac{x - a}{2} \right)}{\sqrt{x} - \sqrt{a}}\]
\[\text{ Dividing the numerator and the denominator by }\sqrt{x} + \sqrt{a}: \]
\[ - 2 \lim_{x \to a} \frac{\left( \sin \left( \frac{x + a}{2} \right)\sin \left( \frac{x - a}{2} \right) \right)}{2\left( \frac{x - a}{2} \right)}\left( \sqrt{x} + \sqrt{a} \right)\]
\[ = - 2 \lim_{x \to a} \sin \left( \frac{x + a}{2} \right) \times \frac{\sin \left( \frac{x - a}{2} \right)}{2\left( \frac{x - a}{2} \right)}\left( \sqrt{x} + \sqrt{a} \right)\]
\[ \Rightarrow - 2 \sin \left( \frac{2a}{2} \right) \left( \frac{\sqrt{a} + \sqrt{a}}{2} \right)\]
\[ = - 2\sqrt{a} \sin a\]
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