How to solve a $\lim_{x\to 0}\frac{\sqrt{1-\cos2x}}{x}$?

Question

This is a home work problem. How to solve:

$$\lim_{x\rightarrow 0} \frac{\sqrt{1-\cos2x}}{x}$$

If I plug $$x=0$$ in the equation I get $$\frac{0}{0}$$

Any ideas how to find this limit? Tips welcome. :)

I know you can take derivative when you've 0 over 0, but in this case it doesn't simplify it. Actually in this homework I can't take derivative and have to solve it somehow else...

Answer 1

HINT:

As $\cos2x=1-2\sin^2x,\sqrt{1-\cos2x}=\sqrt2|\sin x|$

Now for real $a, |a|=+a$ if $a\ge0$ else $|a|=-a$

So, $\lim_{x\to0^-}\dfrac{|\sin x|}x=-\lim_{x\to0^-}\dfrac{\sin x}x=-1$ and you should complete the rest