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Question

Evaluate:
$\lim\limits_{\text{x} \rightarrow{\text{a}}}\frac{\sin\text{x}-\sin\text{a}}{\sqrt{\text{x}}-\sqrt{\text{a}}}$

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Answer

Given that $\lim\limits_{\text{x} \rightarrow{\text{a}}}\frac{\sin\text{x}-\sin\text{a}}{\sqrt{\text{x}}-\sqrt{\text{a}}}$
$=\lim\limits_{\text{x} \rightarrow{\text{a}}}\frac{\sin\text{x}-\sin\text{a}}{\sqrt{\text{x}}-\sqrt{\text{a}}}\times\frac{\sqrt{\text{x}}+\sqrt{\text{a}}}{\sqrt{\text{x}}+\sqrt{\text{a}}}$
$=\lim\limits_{\text{x} \rightarrow{\text{a}}}\frac{\big(2\cos\frac{\text{x}+\text{a}}{2}.\sin\frac{\text{x}-\text{a}}{2}\big)\sqrt{\text{x}}+\sqrt{\text{a}}}{\text{x}-\text{a}}$
$=\lim\limits_{\text{x} \rightarrow{\text{a}}}\cos\big(\frac{\text{x}+\text{a}}{2}\big)\big(\sqrt{\text{x}}+\sqrt{\text{a}}\big)$
Taking limits we have
$=\cos\big(\frac{\text{a}+\text{a}}{2}\big)\big(\sqrt{\text{a}}+\sqrt{\text{a}}\big)$
$=\cos\text{x}\times2\sqrt{\text{a}}=2\sqrt{\text{a}}.\cos\text{a}$
Hence, the required answer is $2\sqrt{\text{a}}.\cos\text{a}.$

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