Tenemos la indeterminación de tipo
0/0,
tal que el límite para el numerador es
$$\lim_{x \to 0^+} \frac{1}{\log{\left(\sin{\left(x \right)} \right)}} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\log{\left(\sin{\left(5 x \right)} \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(5 x \right)} \right)}}{\log{\left(\sin{\left(x \right)} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(x \right)} \right)}}}{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(5 x \right)} \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(5 x \right)} \right)}^{2} \sin{\left(5 x \right)} \cos{\left(x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)} \cos{\left(5 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(5 x \right)} \right)}^{2} \sin{\left(5 x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{\sin{\left(5 x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}}}{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(5 x \right)} \right)}^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(\frac{\cos{\left(5 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}} - \frac{\sin{\left(5 x \right)} \cos{\left(x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin^{2}{\left(x \right)}} - \frac{2 \sin{\left(5 x \right)} \cos{\left(x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{3} \sin^{2}{\left(x \right)}}\right) \log{\left(\sin{\left(5 x \right)} \right)}^{3} \sin{\left(5 x \right)}}{10 \cos{\left(5 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(\frac{\cos{\left(5 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}} - \frac{\sin{\left(5 x \right)} \cos{\left(x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin^{2}{\left(x \right)}} - \frac{2 \sin{\left(5 x \right)} \cos{\left(x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{3} \sin^{2}{\left(x \right)}}\right) \log{\left(\sin{\left(5 x \right)} \right)}^{3} \sin{\left(5 x \right)}}{10}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{\log{\left(\sin{\left(5 x \right)} \right)}^{3} \sin{\left(5 x \right)}}{10}\right)}{\frac{d}{d x} \frac{1}{\frac{\cos{\left(5 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}} - \frac{\sin{\left(5 x \right)} \cos{\left(x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin^{2}{\left(x \right)}} - \frac{2 \sin{\left(5 x \right)} \cos{\left(x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{3} \sin^{2}{\left(x \right)}}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{\log{\left(\sin{\left(5 x \right)} \right)}^{3} \cos^{3}{\left(5 x \right)}}{2 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{2}{\left(x \right)}} + \frac{\log{\left(\sin{\left(5 x \right)} \right)}^{3} \sin{\left(5 x \right)} \cos{\left(x \right)} \cos^{2}{\left(5 x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{3}{\left(x \right)}} - \frac{\log{\left(\sin{\left(5 x \right)} \right)}^{3} \sin^{2}{\left(5 x \right)} \cos^{2}{\left(x \right)} \cos{\left(5 x \right)}}{50 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{4}{\left(x \right)}} - \frac{3 \log{\left(\sin{\left(5 x \right)} \right)}^{2} \cos^{3}{\left(5 x \right)}}{2 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{2}{\left(x \right)}} + \frac{3 \log{\left(\sin{\left(5 x \right)} \right)}^{2} \sin{\left(5 x \right)} \cos{\left(x \right)} \cos^{2}{\left(5 x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{3}{\left(x \right)}} - \frac{3 \log{\left(\sin{\left(5 x \right)} \right)}^{2} \sin^{2}{\left(5 x \right)} \cos^{2}{\left(x \right)} \cos{\left(5 x \right)}}{50 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{4}{\left(x \right)}} + \frac{2 \log{\left(\sin{\left(5 x \right)} \right)}^{3} \sin{\left(5 x \right)} \cos{\left(x \right)} \cos^{2}{\left(5 x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{5} \sin^{3}{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(5 x \right)} \right)}^{3} \sin^{2}{\left(5 x \right)} \cos^{2}{\left(x \right)} \cos{\left(5 x \right)}}{25 \log{\left(\sin{\left(x \right)} \right)}^{5} \sin^{4}{\left(x \right)}} + \frac{6 \log{\left(\sin{\left(5 x \right)} \right)}^{2} \sin{\left(5 x \right)} \cos{\left(x \right)} \cos^{2}{\left(5 x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{5} \sin^{3}{\left(x \right)}} - \frac{6 \log{\left(\sin{\left(5 x \right)} \right)}^{2} \sin^{2}{\left(5 x \right)} \cos^{2}{\left(x \right)} \cos{\left(5 x \right)}}{25 \log{\left(\sin{\left(x \right)} \right)}^{5} \sin^{4}{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(5 x \right)} \right)}^{3} \sin^{2}{\left(5 x \right)} \cos^{2}{\left(x \right)} \cos{\left(5 x \right)}}{25 \log{\left(\sin{\left(x \right)} \right)}^{6} \sin^{4}{\left(x \right)}} - \frac{6 \log{\left(\sin{\left(5 x \right)} \right)}^{2} \sin^{2}{\left(5 x \right)} \cos^{2}{\left(x \right)} \cos{\left(5 x \right)}}{25 \log{\left(\sin{\left(x \right)} \right)}^{6} \sin^{4}{\left(x \right)}}}{\frac{24 \sin{\left(5 x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}} + \frac{2 \cos{\left(x \right)} \cos{\left(5 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{2} \sin^{2}{\left(x \right)}} - \frac{2 \sin{\left(5 x \right)} \cos^{2}{\left(x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin^{3}{\left(x \right)}} - \frac{2 \sin{\left(5 x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{3} \sin{\left(x \right)}} + \frac{4 \cos{\left(x \right)} \cos{\left(5 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{3} \sin^{2}{\left(x \right)}} - \frac{6 \sin{\left(5 x \right)} \cos^{2}{\left(x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{3} \sin^{3}{\left(x \right)}} - \frac{6 \sin{\left(5 x \right)} \cos^{2}{\left(x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{3}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{\log{\left(\sin{\left(5 x \right)} \right)}^{3} \cos^{3}{\left(5 x \right)}}{2 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{2}{\left(x \right)}} + \frac{\log{\left(\sin{\left(5 x \right)} \right)}^{3} \sin{\left(5 x \right)} \cos{\left(x \right)} \cos^{2}{\left(5 x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{3}{\left(x \right)}} - \frac{\log{\left(\sin{\left(5 x \right)} \right)}^{3} \sin^{2}{\left(5 x \right)} \cos^{2}{\left(x \right)} \cos{\left(5 x \right)}}{50 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{4}{\left(x \right)}} - \frac{3 \log{\left(\sin{\left(5 x \right)} \right)}^{2} \cos^{3}{\left(5 x \right)}}{2 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{2}{\left(x \right)}} + \frac{3 \log{\left(\sin{\left(5 x \right)} \right)}^{2} \sin{\left(5 x \right)} \cos{\left(x \right)} \cos^{2}{\left(5 x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{3}{\left(x \right)}} - \frac{3 \log{\left(\sin{\left(5 x \right)} \right)}^{2} \sin^{2}{\left(5 x \right)} \cos^{2}{\left(x \right)} \cos{\left(5 x \right)}}{50 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{4}{\left(x \right)}} + \frac{2 \log{\left(\sin{\left(5 x \right)} \right)}^{3} \sin{\left(5 x \right)} \cos{\left(x \right)} \cos^{2}{\left(5 x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{5} \sin^{3}{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(5 x \right)} \right)}^{3} \sin^{2}{\left(5 x \right)} \cos^{2}{\left(x \right)} \cos{\left(5 x \right)}}{25 \log{\left(\sin{\left(x \right)} \right)}^{5} \sin^{4}{\left(x \right)}} + \frac{6 \log{\left(\sin{\left(5 x \right)} \right)}^{2} \sin{\left(5 x \right)} \cos{\left(x \right)} \cos^{2}{\left(5 x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{5} \sin^{3}{\left(x \right)}} - \frac{6 \log{\left(\sin{\left(5 x \right)} \right)}^{2} \sin^{2}{\left(5 x \right)} \cos^{2}{\left(x \right)} \cos{\left(5 x \right)}}{25 \log{\left(\sin{\left(x \right)} \right)}^{5} \sin^{4}{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(5 x \right)} \right)}^{3} \sin^{2}{\left(5 x \right)} \cos^{2}{\left(x \right)} \cos{\left(5 x \right)}}{25 \log{\left(\sin{\left(x \right)} \right)}^{6} \sin^{4}{\left(x \right)}} - \frac{6 \log{\left(\sin{\left(5 x \right)} \right)}^{2} \sin^{2}{\left(5 x \right)} \cos^{2}{\left(x \right)} \cos{\left(5 x \right)}}{25 \log{\left(\sin{\left(x \right)} \right)}^{6} \sin^{4}{\left(x \right)}}}{\frac{24 \sin{\left(5 x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}} + \frac{2 \cos{\left(x \right)} \cos{\left(5 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{2} \sin^{2}{\left(x \right)}} - \frac{2 \sin{\left(5 x \right)} \cos^{2}{\left(x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin^{3}{\left(x \right)}} - \frac{2 \sin{\left(5 x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{3} \sin{\left(x \right)}} + \frac{4 \cos{\left(x \right)} \cos{\left(5 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{3} \sin^{2}{\left(x \right)}} - \frac{6 \sin{\left(5 x \right)} \cos^{2}{\left(x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{3} \sin^{3}{\left(x \right)}} - \frac{6 \sin{\left(5 x \right)} \cos^{2}{\left(x \right)}}{5 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{3}{\left(x \right)}}}\right)$$
=
$$1$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)