Evaluate the following limits:
Question:

Evaluate the following limits:

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

 

Solution:

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

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

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

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

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

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

$=1$

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

 

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