Límite de la función log(x)*log(acos(x))

Tenemos la indeterminación de tipo

0/0,

tal que el límite para el numerador es
$$\lim_{x \to 1^+} \log{\left(x \right)} = 0$$
y el límite para el denominador es
$$\lim_{x \to 1^+} \frac{1}{\log{\left(\operatorname{acos}{\left(x \right)} \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 1^+}\left(\log{\left(x \right)} \log{\left(\operatorname{acos}{\left(x \right)} \right)}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \log{\left(x \right)}}{\frac{d}{d x} \frac{1}{\log{\left(\operatorname{acos}{\left(x \right)} \right)}}}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\sqrt{1 - x^{2}} \log{\left(\operatorname{acos}{\left(x \right)} \right)}^{2} \operatorname{acos}{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \frac{\log{\left(\operatorname{acos}{\left(x \right)} \right)}^{2} \operatorname{acos}{\left(x \right)}}{x}}{\frac{d}{d x} \frac{1}{\sqrt{1 - x^{2}}}}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\left(1 - x^{2}\right)^{\frac{3}{2}} \left(- \frac{\log{\left(\operatorname{acos}{\left(x \right)} \right)}^{2}}{x \sqrt{1 - x^{2}}} - \frac{2 \log{\left(\operatorname{acos}{\left(x \right)} \right)}}{x \sqrt{1 - x^{2}}} - \frac{\log{\left(\operatorname{acos}{\left(x \right)} \right)}^{2} \operatorname{acos}{\left(x \right)}}{x^{2}}\right)}{x}\right)$$
=
$$\lim_{x \to 1^+}\left(\left(1 - x^{2}\right)^{\frac{3}{2}} \left(- \frac{\log{\left(\operatorname{acos}{\left(x \right)} \right)}^{2}}{x \sqrt{1 - x^{2}}} - \frac{2 \log{\left(\operatorname{acos}{\left(x \right)} \right)}}{x \sqrt{1 - x^{2}}} - \frac{\log{\left(\operatorname{acos}{\left(x \right)} \right)}^{2} \operatorname{acos}{\left(x \right)}}{x^{2}}\right)\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(1 - x^{2}\right)^{\frac{3}{2}}}{\frac{d}{d x} \frac{1}{- \frac{\log{\left(\operatorname{acos}{\left(x \right)} \right)}^{2}}{x \sqrt{1 - x^{2}}} - \frac{2 \log{\left(\operatorname{acos}{\left(x \right)} \right)}}{x \sqrt{1 - x^{2}}} - \frac{\log{\left(\operatorname{acos}{\left(x \right)} \right)}^{2} \operatorname{acos}{\left(x \right)}}{x^{2}}}}\right)$$
=
$$\lim_{x \to 1^+}\left(- \frac{3 x \sqrt{1 - x^{2}} \left(- \frac{\log{\left(\operatorname{acos}{\left(x \right)} \right)}^{2}}{x \sqrt{1 - x^{2}}} - \frac{2 \log{\left(\operatorname{acos}{\left(x \right)} \right)}}{x \sqrt{1 - x^{2}}} - \frac{\log{\left(\operatorname{acos}{\left(x \right)} \right)}^{2} \operatorname{acos}{\left(x \right)}}{x^{2}}\right)^{2}}{\frac{\log{\left(\operatorname{acos}{\left(x \right)} \right)}^{2}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{2 \log{\left(\operatorname{acos}{\left(x \right)} \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} - \frac{2 \log{\left(\operatorname{acos}{\left(x \right)} \right)}}{x \left(1 - x^{2}\right) \operatorname{acos}{\left(x \right)}} - \frac{2}{x \left(1 - x^{2}\right) \operatorname{acos}{\left(x \right)}} - \frac{2 \log{\left(\operatorname{acos}{\left(x \right)} \right)}^{2}}{x^{2} \sqrt{1 - x^{2}}} - \frac{4 \log{\left(\operatorname{acos}{\left(x \right)} \right)}}{x^{2} \sqrt{1 - x^{2}}} - \frac{2 \log{\left(\operatorname{acos}{\left(x \right)} \right)}^{2} \operatorname{acos}{\left(x \right)}}{x^{3}}}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{- \frac{3 \sqrt{1 - x^{2}} \log{\left(\operatorname{acos}{\left(x \right)} \right)}^{4}}{- x^{4} + x^{2}} - \frac{12 \sqrt{1 - x^{2}} \log{\left(\operatorname{acos}{\left(x \right)} \right)}^{3}}{- x^{4} + x^{2}} - \frac{12 \sqrt{1 - x^{2}} \log{\left(\operatorname{acos}{\left(x \right)} \right)}^{2}}{- x^{4} + x^{2}} - \frac{6 \log{\left(\operatorname{acos}{\left(x \right)} \right)}^{4} \operatorname{acos}{\left(x \right)}}{x^{3}} - \frac{12 \log{\left(\operatorname{acos}{\left(x \right)} \right)}^{3} \operatorname{acos}{\left(x \right)}}{x^{3}} - \frac{3 \sqrt{1 - x^{2}} \log{\left(\operatorname{acos}{\left(x \right)} \right)}^{4} \operatorname{acos}^{2}{\left(x \right)}}{x^{4}}}{- \frac{2 \log{\left(\operatorname{acos}{\left(x \right)} \right)}}{- x^{3} \operatorname{acos}{\left(x \right)} + x \operatorname{acos}{\left(x \right)}} - \frac{2}{- x^{3} \operatorname{acos}{\left(x \right)} + x \operatorname{acos}{\left(x \right)}} + \frac{\log{\left(\operatorname{acos}{\left(x \right)} \right)}^{2}}{- x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}} + \frac{2 \log{\left(\operatorname{acos}{\left(x \right)} \right)}}{- x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}} - \frac{2 \log{\left(\operatorname{acos}{\left(x \right)} \right)}^{2}}{x^{2} \sqrt{1 - x^{2}}} - \frac{4 \log{\left(\operatorname{acos}{\left(x \right)} \right)}}{x^{2} \sqrt{1 - x^{2}}} - \frac{2 \log{\left(\operatorname{acos}{\left(x \right)} \right)}^{2} \operatorname{acos}{\left(x \right)}}{x^{3}}}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{- \frac{3 \sqrt{1 - x^{2}} \log{\left(\operatorname{acos}{\left(x \right)} \right)}^{4}}{- x^{4} + x^{2}} - \frac{12 \sqrt{1 - x^{2}} \log{\left(\operatorname{acos}{\left(x \right)} \right)}^{3}}{- x^{4} + x^{2}} - \frac{12 \sqrt{1 - x^{2}} \log{\left(\operatorname{acos}{\left(x \right)} \right)}^{2}}{- x^{4} + x^{2}} - \frac{6 \log{\left(\operatorname{acos}{\left(x \right)} \right)}^{4} \operatorname{acos}{\left(x \right)}}{x^{3}} - \frac{12 \log{\left(\operatorname{acos}{\left(x \right)} \right)}^{3} \operatorname{acos}{\left(x \right)}}{x^{3}} - \frac{3 \sqrt{1 - x^{2}} \log{\left(\operatorname{acos}{\left(x \right)} \right)}^{4} \operatorname{acos}^{2}{\left(x \right)}}{x^{4}}}{- \frac{2 \log{\left(\operatorname{acos}{\left(x \right)} \right)}}{- x^{3} \operatorname{acos}{\left(x \right)} + x \operatorname{acos}{\left(x \right)}} - \frac{2}{- x^{3} \operatorname{acos}{\left(x \right)} + x \operatorname{acos}{\left(x \right)}} + \frac{\log{\left(\operatorname{acos}{\left(x \right)} \right)}^{2}}{- x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}} + \frac{2 \log{\left(\operatorname{acos}{\left(x \right)} \right)}}{- x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}} - \frac{2 \log{\left(\operatorname{acos}{\left(x \right)} \right)}^{2}}{x^{2} \sqrt{1 - x^{2}}} - \frac{4 \log{\left(\operatorname{acos}{\left(x \right)} \right)}}{x^{2} \sqrt{1 - x^{2}}} - \frac{2 \log{\left(\operatorname{acos}{\left(x \right)} \right)}^{2} \operatorname{acos}{\left(x \right)}}{x^{3}}}\right)$$
=
$$0$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)