Evaluate the following limits


Evaluate the following limits.

$\lim _{x \rightarrow 0} \frac{(\sqrt{1+2 x}-\sqrt{1-2 x}}{\sin x}$



$=\lim _{x \rightarrow 0} \frac{(\sqrt{1+2 x}-\sqrt{1-2 x})}{\sin x}$

$=\lim _{x \rightarrow 0} \frac{(\sqrt{1+2 x}-\sqrt{1-2 x})}{\sin x} \times \frac{\sqrt{1+2 x}+\sqrt{1-2 x}}{(\sqrt{1+2 x}+\sqrt{1-2 x})}$

$=\lim _{x \rightarrow 0} \frac{1+2 x-1+2 x}{\sin x} \times \frac{1}{\sqrt{1+2 x}+\sqrt{1-2 x}}$

$=4 \times \lim _{x \rightarrow 0} \frac{x}{\sin x} \times \frac{1}{\sqrt{1+2 x}+\sqrt{1-2 x}}$

$=4 \times \frac{1}{2} \times 1$


$\therefore \lim _{x \rightarrow 0} \frac{(\sqrt{1+2 x}-\sqrt{1-2 x})}{\sin x}=2$


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