Question:
Evaluate the following limits.
$\lim _{x \rightarrow 0} \frac{(\sqrt{1+2 x}-\sqrt{1-2 x}}{\sin x}$
Solution:
$=\lim _{x \rightarrow 0} \frac{(\sqrt{1+2 x}-\sqrt{1-2 x})}{\sin x}$
$=\lim _{x \rightarrow 0} \frac{(\sqrt{1+2 x}-\sqrt{1-2 x})}{\sin x} \times \frac{\sqrt{1+2 x}+\sqrt{1-2 x}}{(\sqrt{1+2 x}+\sqrt{1-2 x})}$
$=\lim _{x \rightarrow 0} \frac{1+2 x-1+2 x}{\sin x} \times \frac{1}{\sqrt{1+2 x}+\sqrt{1-2 x}}$
$=4 \times \lim _{x \rightarrow 0} \frac{x}{\sin x} \times \frac{1}{\sqrt{1+2 x}+\sqrt{1-2 x}}$
$=4 \times \frac{1}{2} \times 1$
$=2$
$\therefore \lim _{x \rightarrow 0} \frac{(\sqrt{1+2 x}-\sqrt{1-2 x})}{\sin x}=2$
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