Question

The value of $\underset{x\to 0}{lim}\frac{\sqrt{1-cos2x}}{x}equals$

• A. 0
• B. 1
• C. $\sqrt{2}$
• D. Does not exist

Answer 1

The correct option is C $\sqrt{2}$

$\underset{x\to 0}{lim}\frac{\sqrt{1-\cos 2x}}{x}$

We know $1-\cos 2x=2{\sin }^{2}x$

$\Rightarrow$ $\underset{x\to 0}{lim}\frac{\sqrt{2{\sin }^{2}x}}{x}$

$\Rightarrow$ $\underset{x\to 0}{lim}\frac{\sqrt{2}\sin x}{x}$

$\Rightarrow$ $\sqrt{2}\underset{x\to 0}{lim}\frac{\sin x}{x}$

$\Rightarrow$ $\sqrt{2}\left(1\right)$

$\Rightarrow$ $\sqrt{2}$

$\therefore$ $\underset{x\to 0}{lim}\frac{\sqrt{1-\cos 2x}}{x}=\sqrt{2}$