Evaluate the following limits:

Question:

Evaluate the following limits:

$\lim _{x \rightarrow a} \frac{(\sin x-\sin a)}{(\sqrt{x}-\sqrt{a})}$

 

Solution:

$=\lim _{x \rightarrow a} \frac{(\sin x-\sin a)}{(\sqrt{x}-\sqrt{a})}$

$=\lim _{x \rightarrow a} \frac{(\sin x-\sin a)}{(\sqrt{x}-\sqrt{a})} \times \frac{(\sqrt{x}+\sqrt{a})}{(\sqrt{x}+\sqrt{a})}[$ Multiply and divide by $\sqrt{x}-\sqrt{a}]$

$=\lim _{x \rightarrow a} \frac{(\sin x-\sin a) \times(\sqrt{x}+\sqrt{a})}{(x-a)}$

$=\cos a \times \lim _{x \rightarrow a}(\sqrt{x}+\sqrt{a})\left[\because \lim _{x \rightarrow a} \frac{\sin x-\sin a}{x-a}=\cos a\right]$

$=2 \sqrt{a} \times \cos a$

$=2 \sqrt{a c o s a}$

$\therefore \lim _{x \rightarrow a} \frac{(\sin x-\sin a)}{(\sqrt{x}-\sqrt{a})}=2 \sqrt{a} \cos a$

 

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