Límite de la función sqrt(x)*log(sin(x))

Tenemos la indeterminación de tipo

0/0,

tal que el límite para el numerador es
$$\lim_{x \to 0^+} \sqrt{x} = 0$$
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.
$$\lim_{x \to 0^+}\left(\sqrt{x} \log{\left(\sin{\left(x \right)} \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sqrt{x}}{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(x \right)} \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}}{2 \sqrt{x} \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}}{2 \sqrt{x}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{\sin{\left(x \right)}}{2 \sqrt{x}}\right)}{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(x \right)} \right)}^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(- \frac{\cos{\left(x \right)}}{2 \sqrt{x}} + \frac{\sin{\left(x \right)}}{4 x^{\frac{3}{2}}}\right) \log{\left(\sin{\left(x \right)} \right)}^{3} \sin{\left(x \right)}}{2 \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(- \frac{\cos{\left(x \right)}}{2 \sqrt{x}} + \frac{\sin{\left(x \right)}}{4 x^{\frac{3}{2}}}\right) \log{\left(\sin{\left(x \right)} \right)}^{3} \sin{\left(x \right)}}{2}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{\log{\left(\sin{\left(x \right)} \right)}^{3} \sin{\left(x \right)}}{2}\right)}{\frac{d}{d x} \frac{1}{- \frac{\cos{\left(x \right)}}{2 \sqrt{x}} + \frac{\sin{\left(x \right)}}{4 x^{\frac{3}{2}}}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{\log{\left(\sin{\left(x \right)} \right)}^{3} \cos^{3}{\left(x \right)}}{8 x} - \frac{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \cos^{3}{\left(x \right)}}{8 x} + \frac{\log{\left(\sin{\left(x \right)} \right)}^{3} \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{8 x^{2}} + \frac{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{8 x^{2}} - \frac{\log{\left(\sin{\left(x \right)} \right)}^{3} \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{32 x^{3}} - \frac{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{32 x^{3}}}{- \frac{\sin{\left(x \right)}}{2 \sqrt{x}} - \frac{\cos{\left(x \right)}}{2 x^{\frac{3}{2}}} + \frac{3 \sin{\left(x \right)}}{8 x^{\frac{5}{2}}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{\log{\left(\sin{\left(x \right)} \right)}^{3} \cos^{3}{\left(x \right)}}{8 x} - \frac{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \cos^{3}{\left(x \right)}}{8 x} + \frac{\log{\left(\sin{\left(x \right)} \right)}^{3} \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{8 x^{2}} + \frac{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{8 x^{2}} - \frac{\log{\left(\sin{\left(x \right)} \right)}^{3} \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{32 x^{3}} - \frac{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{32 x^{3}}}{- \frac{\sin{\left(x \right)}}{2 \sqrt{x}} - \frac{\cos{\left(x \right)}}{2 x^{\frac{3}{2}}} + \frac{3 \sin{\left(x \right)}}{8 x^{\frac{5}{2}}}}\right)$$
=
$$0$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)