Limits-2

For small $x$ , $\cos x \approx 1 - \frac {x^2}{2}$ , and $(1 + x)^n \approx 1 + nx$

$\begin{aligned} \displaystyle \lim_{x \to 0} \frac {1 - \cos x \sqrt{ \cos 2x } }{x^2} & = & \lim_{x \to 0} \frac {1 - \left (1 - \frac {x^2}{2} \right ) \sqrt{1 - 2x^2}}{x^2} \\ \displaystyle & = & \lim_{x \to 0} \frac {2 - (2-x^2)(1 - x^2)}{2x^2} \\ \displaystyle & = & \lim_{x \to 0} \frac {3-x^2}{2} = \boxed{1.5} \\ \end{aligned}$

Applied L'Hospital Rule twice, and got 1/2.