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

Question

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

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Answer

$\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\cos\text{x}-\cos\text{a}}{\sqrt{\text{x}}-\sqrt{\text{a}}}$ $=\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\Big(-2\sin\big(\frac{\text{x}+\text{a}}{2}\big)\sin\big(\frac{\text{x}-\text{a}}{2}\big)\Big)\times\big(\sqrt{\text{x}}+\sqrt{\text{a}}\big)}{\big(\sqrt{\text{x}}-\sqrt{\text{a}}\big)\big(\sqrt{\text{x}}+\sqrt{\text{a}}\big)}$ $=-2\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\sin\big(\frac{\text{x}+\text{a}}{2}\big)\sin\big(\frac{\text{x}-\text{a}}{2}\big)\times\big(\sqrt{\text{x}}+\sqrt{\text{a}}\big)}{(\text{x}-\text{a})}$ $=-2\lim\limits_{\text{x}\rightarrow{\text{a}}}{\sin\big(\frac{\text{x}+\text{a}}{2}\big)}\times\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\sin\big(\frac{\text{x}-\text{a}}{2}\big)\times\frac12}{\big(\frac{\text{x}-\text{a}}{2}\big)}\lim\limits_{\text{x}\rightarrow{\text{a}}}\big(\sqrt{\text{x}}+\sqrt{\text{a}}\big)$ $=-2\times\sin(\text{a})\times1\times\frac12\times2\sqrt{\text{a}}$ $=-2\sqrt{\text{a}}\sin\text{a}$

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