lim_{x\rightarrow 0} \frac{1-\cos x}{x^2} =lim_{x\rightarrow 0} \frac{(1-\cos x)(1+\cos x)}{2x^2}=lim_{x\rightarrow 0} \frac{1-\cos^2x}{2x^2}=lim_{x\rightarrow 0} \frac{\sin^2 x}{2x^2}=\frac{1}{2}