Tenemos la indeterminación de tipo
0/0,
tal que el límite para el numerador es
$$\lim_{x \to 0^+} \sin{\left(x \right)} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\log{\left(\sin{\left(x \right)} \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(x \right)}}{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(x \right)} \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \sin{\left(x \right)} \cos{\left(x \right)}\right)}{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(x \right)} \right)}^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} \right)}^{3} \sin{\left(x \right)}}{2 \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} \right)}^{3} \sin{\left(x \right)}}{2}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{\log{\left(\sin{\left(x \right)} \right)}^{3} \sin{\left(x \right)}}{2}\right)}{\frac{d}{d x} \frac{1}{\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(- \frac{\log{\left(\sin{\left(x \right)} \right)}^{3} \cos{\left(x \right)}}{2} - \frac{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \cos{\left(x \right)}}{2}\right) \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)^{2}}{4 \sin{\left(x \right)} \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{- \frac{\log{\left(\sin{\left(x \right)} \right)}^{3} \cos{\left(x \right)}}{2} - \frac{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \cos{\left(x \right)}}{2}}{4 \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{- \frac{\log{\left(\sin{\left(x \right)} \right)}^{3} \cos{\left(x \right)}}{2} - \frac{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \cos{\left(x \right)}}{2}}{4 \sin{\left(x \right)}}\right)$$
=
$$0$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)