Tenemos la indeterminación de tipo
0/0,
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(- \cos{\left(x \right)} \sqrt{\cos{\left(2 x \right)}} + 1\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \tan^{2}{\left(x \right)} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{- \cos{\left(x \right)} \sqrt{\cos{\left(2 x \right)}} + 1}{\tan^{2}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \cos{\left(x \right)} \sqrt{\cos{\left(2 x \right)}} + 1\right)}{\frac{d}{d x} \tan^{2}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} \sqrt{\cos{\left(2 x \right)}} + \frac{\sin{\left(2 x \right)} \cos{\left(x \right)}}{\sqrt{\cos{\left(2 x \right)}}}}{\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} \sqrt{\cos{\left(2 x \right)}} + \frac{\sin{\left(2 x \right)} \cos{\left(x \right)}}{\sqrt{\cos{\left(2 x \right)}}}}{2 \tan{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\sin{\left(x \right)} \sqrt{\cos{\left(2 x \right)}} + \frac{\sin{\left(2 x \right)} \cos{\left(x \right)}}{\sqrt{\cos{\left(2 x \right)}}}\right)}{\frac{d}{d x} 2 \tan{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{2 \sin{\left(x \right)} \sin{\left(2 x \right)}}{\sqrt{\cos{\left(2 x \right)}}} + \frac{\sin^{2}{\left(2 x \right)} \cos{\left(x \right)}}{\cos^{\frac{3}{2}}{\left(2 x \right)}} + 3 \cos{\left(x \right)} \sqrt{\cos{\left(2 x \right)}}}{2 \tan^{2}{\left(x \right)} + 2}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{2 \sin{\left(x \right)} \sin{\left(2 x \right)}}{\sqrt{\cos{\left(2 x \right)}}} + \frac{\sin^{2}{\left(2 x \right)} \cos{\left(x \right)}}{\cos^{\frac{3}{2}}{\left(2 x \right)}} + 3 \cos{\left(x \right)} \sqrt{\cos{\left(2 x \right)}}}{2 \tan^{2}{\left(x \right)} + 2}\right)$$
=
$$\frac{3}{2}$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 2 vez (veces)