Tenemos la indeterminación de tipo
0/0,
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(x^{2} \cos^{2}{\left(x \right)} - \sin^{2}{\left(x \right)}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+}\left(x^{2} \sin^{2}{\left(x \right)}\right) = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{1}{x^{2}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\frac{x^{2} \cos^{2}{\left(x \right)} - \sin^{2}{\left(x \right)}}{x^{2} \sin^{2}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(x^{2} \cos^{2}{\left(x \right)} - \sin^{2}{\left(x \right)}\right)}{\frac{d}{d x} x^{2} \sin^{2}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- 2 x^{2} \sin{\left(x \right)} \cos{\left(x \right)} + 2 x \cos^{2}{\left(x \right)} - 2 \sin{\left(x \right)} \cos{\left(x \right)}}{2 x^{2} \sin{\left(x \right)} \cos{\left(x \right)} + 2 x \sin^{2}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- 2 x^{2} \sin{\left(x \right)} \cos{\left(x \right)} + 2 x \cos^{2}{\left(x \right)} - 2 \sin{\left(x \right)} \cos{\left(x \right)}\right)}{\frac{d}{d x} \left(2 x^{2} \sin{\left(x \right)} \cos{\left(x \right)} + 2 x \sin^{2}{\left(x \right)}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 x^{2} \sin^{2}{\left(x \right)} - 2 x^{2} \cos^{2}{\left(x \right)} - 8 x \sin{\left(x \right)} \cos{\left(x \right)} + 2 \sin^{2}{\left(x \right)}}{- 2 x^{2} \sin^{2}{\left(x \right)} + 2 x^{2} \cos^{2}{\left(x \right)} + 8 x \sin{\left(x \right)} \cos{\left(x \right)} + 2 \sin^{2}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 x^{2} \sin^{2}{\left(x \right)} - 2 x^{2} \cos^{2}{\left(x \right)} - 8 x \sin{\left(x \right)} \cos{\left(x \right)} + 2 \sin^{2}{\left(x \right)}}{- 2 x^{2} \sin^{2}{\left(x \right)} + 2 x^{2} \cos^{2}{\left(x \right)} + 8 x \sin{\left(x \right)} \cos{\left(x \right)} + 2 \sin^{2}{\left(x \right)}}\right)$$
=
$$- \frac{2}{3}$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 2 vez (veces)