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