I've been asked to find the limit, without using L'hospital's rule, of: $$\lim\limits_{x\to0}\frac{\sqrt{1-\cos(2x)}}{x}$$
Here's my attempt: $$\begin{aligned}&\lim\limits_{x\to0}\frac{\sqrt{1-\cos(2x)}}{x}\\=&\lim_{x\to0}\frac{\sqrt{2\sin^2(x)}}{x}\\=&\lim_{x\to0}\frac{\sqrt{2}\sin(x)}{x}\\=&\sqrt{2}\end{aligned}$$
So my question is what's the problem here? The graph shows that limit doesn't exist. In which situations do we have to find LHL and RHL?