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