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

Question

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

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Answer

$\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\sin\sqrt{\text{x}}-\sin\sqrt{\text{a}}}{\text{x}-\text{a}}$$=\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\sin\sqrt{\text{x}}-\sin\sqrt{\text{a}}}{\big(\sqrt{\text{x}}-\sqrt{\text{a}}\big)\big(\sqrt{\text{x}}+\sqrt{\text{a}}\big)}$
$=\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{2\sin\Big(\frac{\sqrt{\text{x}}-\sqrt{\text{a}}}{2}\Big)\cos\Big(\frac{\sqrt{\text{x}}+\sqrt{\text{a}}}{2}\Big)}{\big(\sqrt{\text{x}}+\sqrt{\text{a}}\big)\Big({\sqrt{\text{x}}-\sqrt{\text{a}}}\Big)}$
$=2\begin{pmatrix}\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\sin\frac{\sqrt{\text{x}}-\sqrt{\text{a}}}{2}}{\Big(\frac{{\sqrt{\text{x}}+\sqrt{\text{a}}}}{2}\Big)}\end{pmatrix}\times\frac12\frac{\lim\limits_{\text{x}\rightarrow{\text{a}}}\cos\Big(\frac{\sqrt{\text{x}}+\sqrt{\text{a}}}{2}\Big)}{\lim\limits_{\text{x}\rightarrow{\text{a}}}\big(\sqrt{\text{x}}+\sqrt{\text{a}}\big)}$
$=2\times1\times\frac{1}{2}\times\cos\sqrt{\text{a}}\times\frac{1}{2\sqrt{\text{a}}}$
$=\frac{\cos\sqrt{\text{a}}}{2\sqrt{\text{a}}}$

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