Question

Solve
$\underset{x\to 0}{lim}\frac{1-\cos x\sqrt{\cos 2x}}{x^2}$

Answer 1

${lim}_{x\to 0}\frac{1-cosx\sqrt{cos2x}}{x^2}$

using L' Hospital rule , ie. differentiating numerator and denominator

${lim}_{x\to 0}\frac{sinx\sqrt{cos2x}-\frac{cosx}{\sqrt{cos2x}\times \frac{1}{2}\left(sin2x\right)\times 2}}{2x}$

when ${lim}_{x\to 0}\frac{sinx}{x}=1$

$\Rightarrow \frac{\sqrt{cos2x}+\frac{sin2xcosx}{\sqrt{cos2x}}}{2}$

$\Rightarrow \frac{1+0}{2}=\frac{1}{2}$