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