Límite de la función pi*x+2*x*atan(x)

Tenemos la indeterminación de tipo

-oo/oo,

tal que el límite para el numerador es
$$\lim_{x \to -\infty} x = -\infty$$
y el límite para el denominador es
$$\lim_{x \to -\infty} \frac{1}{2 \operatorname{atan}{\left(x \right)} + \pi} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to -\infty}\left(\pi x + 2 x \operatorname{atan}{\left(x \right)}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to -\infty}\left(x \left(2 \operatorname{atan}{\left(x \right)} + \pi\right)\right)$$
=
$$\lim_{x \to -\infty}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \frac{1}{2 \operatorname{atan}{\left(x \right)} + \pi}}\right)$$
=
$$\lim_{x \to -\infty}\left(\left(- \frac{x^{2}}{2} - \frac{1}{2}\right) \left(2 \operatorname{atan}{\left(x \right)} + \pi\right)^{2}\right)$$
=
$$\lim_{x \to -\infty}\left(\frac{\frac{d}{d x} \left(- \frac{x^{2}}{2} - \frac{1}{2}\right)}{\frac{d}{d x} \frac{1}{\left(2 \operatorname{atan}{\left(x \right)} + \pi\right)^{2}}}\right)$$
=
$$\lim_{x \to -\infty}\left(\frac{x \left(x^{2} + 1\right) \left(2 \operatorname{atan}{\left(x \right)} + \pi\right)^{3}}{4}\right)$$
=
$$\lim_{x \to -\infty}\left(\frac{\frac{d}{d x} \frac{x \left(x^{2} + 1\right)}{4}}{\frac{d}{d x} \frac{1}{\left(2 \operatorname{atan}{\left(x \right)} + \pi\right)^{3}}}\right)$$
=
$$\lim_{x \to -\infty}\left(- \frac{\left(\frac{3 x^{2}}{4} + \frac{1}{4}\right) \left(x^{2} + 1\right) \left(2 \operatorname{atan}{\left(x \right)} + \pi\right)^{4}}{6}\right)$$
=
$$\lim_{x \to -\infty}\left(\frac{\frac{d}{d x} \left(- \frac{\left(\frac{3 x^{2}}{4} + \frac{1}{4}\right) \left(x^{2} + 1\right)}{6}\right)}{\frac{d}{d x} \frac{1}{\left(2 \operatorname{atan}{\left(x \right)} + \pi\right)^{4}}}\right)$$
=
$$\lim_{x \to -\infty}\left(\left(- \frac{x^{3}}{2} - \frac{x}{3}\right) \left(- 4 x^{2} \operatorname{atan}^{5}{\left(x \right)} - 10 \pi x^{2} \operatorname{atan}^{4}{\left(x \right)} - 10 \pi^{2} x^{2} \operatorname{atan}^{3}{\left(x \right)} - 5 \pi^{3} x^{2} \operatorname{atan}^{2}{\left(x \right)} - \frac{5 \pi^{4} x^{2} \operatorname{atan}{\left(x \right)}}{4} - \frac{\pi^{5} x^{2}}{8} - 4 \operatorname{atan}^{5}{\left(x \right)} - 10 \pi \operatorname{atan}^{4}{\left(x \right)} - 10 \pi^{2} \operatorname{atan}^{3}{\left(x \right)} - 5 \pi^{3} \operatorname{atan}^{2}{\left(x \right)} - \frac{5 \pi^{4} \operatorname{atan}{\left(x \right)}}{4} - \frac{\pi^{5}}{8}\right)\right)$$
=
$$\lim_{x \to -\infty}\left(\left(- \frac{x^{3}}{2} - \frac{x}{3}\right) \left(- 4 x^{2} \operatorname{atan}^{5}{\left(x \right)} - 10 \pi x^{2} \operatorname{atan}^{4}{\left(x \right)} - 10 \pi^{2} x^{2} \operatorname{atan}^{3}{\left(x \right)} - 5 \pi^{3} x^{2} \operatorname{atan}^{2}{\left(x \right)} - \frac{5 \pi^{4} x^{2} \operatorname{atan}{\left(x \right)}}{4} - \frac{\pi^{5} x^{2}}{8} - 4 \operatorname{atan}^{5}{\left(x \right)} - 10 \pi \operatorname{atan}^{4}{\left(x \right)} - 10 \pi^{2} \operatorname{atan}^{3}{\left(x \right)} - 5 \pi^{3} \operatorname{atan}^{2}{\left(x \right)} - \frac{5 \pi^{4} \operatorname{atan}{\left(x \right)}}{4} - \frac{\pi^{5}}{8}\right)\right)$$
=
$$-2$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 4 vez (veces)