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