Evaluate the following limit: _ x a x - a x-a - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\cos\sqrt{\text{x}}-\cos\sqrt{\text{a}}}{\text{x}-\text{a}}$

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Answer

$\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\cos\sqrt{\text{x}}-\cos\sqrt{\text{a}}}{\text{x}-\text{a}}$

$=\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{-2\sin\Big(\frac{\sqrt{\text{x}}+\sqrt{\text{a}}}{2}\Big)\times\sin\Big(\frac{\sqrt{\text{x}}-\sqrt{\text{a}}}{2}\Big)}{\big(\sqrt{\text{x}}+\sqrt{\text{a}}\big)\times\Big({\sqrt{\text{x}}-\sqrt{\text{a}}}\Big)}$

$=-2\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\sin\Big(\frac{\sqrt{\text{x}}+\sqrt{\text{a}}}{2}\Big)\times\lim\limits_{\text{x}\rightarrow{\text{a}}}\sin\Big(\frac{\sqrt{\text{x}}-\sqrt{\text{a}}}{2}\Big)}{\lim\limits_{\text{x}\rightarrow{\text{a}}}\big(\sqrt{\text{x}}+\sqrt{\text{a}}\big)\times\Big(\frac{\sqrt{\text{x}}-\sqrt{\text{a}}}{2}\Big)}\times\frac12$

$=-2\sin\sqrt{\text{a}}\times1\times\frac{1}{2\sqrt{\text{a}}}\times\frac12$

$=-\frac{1}{2\sqrt{\text{a}}}\sin\sqrt{\text{a}}$

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