Limit of $\frac{1-2\cos(x)+\cos^2(2x)}{x^2}$ I tried to find the value of this limit without L'Hopital , but no luck 
$$\lim_{x\to 0}\frac{1-2\cos(x)+\cos^2(2x)}{x^2}$$

$$\frac{1-2\cos(x)+\cos^2(2x)}{x^2}=\frac{4\sin^2\frac{x}{2}-\sin^2(2x)}{x^2}$$
and since $\lim_{x\to 0}\frac{\sin x}{x}=1$, your limit equals $1-4=\color{red}{-3}$.
