Sr Examen
Otras calculadoras:
-
¿Cómo usar?
- Límite de la función:
-
Límite de 1/(1-x)-3/(1-x^3)
-
Límite de sin(3*x)/(2*x)
-
Límite de (1-2*x)^(1/x)
-
Límite de (6+x^2-5*x)/(-9+x^2)
-
Expresiones idénticas
- x*log(dos *atan(x)/pi)
- x multiplicar por logaritmo de (2 multiplicar por arco tangente de gente de (x) dividir por número pi )
- x multiplicar por logaritmo de (dos multiplicar por arco tangente de gente de (x) dividir por número pi )
- xlog(2atan(x)/pi)
- xlog2atanx/pi
- x*log(2*atan(x) dividir por pi)
-
Expresiones semejantes
- x*log(2*arctan(x)/pi)
- x*log(2*arctanx/pi)
-
Expresiones con funciones
- Logaritmo log
- log(x)/log(1+x)
- log(log(x))
- log(sin(m*x))/log(sin(x))
- log(1+3*x)/x
- log(x)/(-1+x^2)
- Arcotangente arctan
- atan(x/2)
- atan(6*x)/(-3*x+2*x^2)
- atan(1/(-1+x))
- atan(2+x)/(2+x)
- atan(x/3)
- Número Pi pi
- Piecewise((x,x<0),(-x,x<2),(0,True))
- pi*x*sin(pi/x)^2
- Piecewise((3+x^2+5*x,x<=-2),(-2+x^2,True))
- pi^4*sin(x)/((pi+x)*(pi-x))
- pi/4-x*atan(x/(1+x))
Límite de la función x*log(2*atan(x)/pi)
Solución
Ha introducido
[src]
/ /2*atan(x)\\ lim |x*log|---------|| x->oo\ \ pi //
$$\lim_{x \to \infty}\left(x \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}\right)$$
Limit(x*log((2*atan(x))/pi), x, oo, dir='-')
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to \infty} x = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} \frac{1}{\log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}} = -\infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(x \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \infty}\left(x \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \frac{1}{\log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \left(x^{2} + 1\right) \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}^{2} \operatorname{atan}{\left(x \right)}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{\pi \left(x^{2} + 1\right) \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}^{2}}{2}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- \frac{\pi \left(x^{2} + 1\right)}{2}\right)}{\frac{d}{d x} \frac{1}{\log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}^{2}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\pi x \left(x^{2} + 1\right) \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}^{3} \operatorname{atan}{\left(x \right)}}{2}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\pi^{2} x \left(x^{2} + 1\right) \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}^{3}}{4}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{\pi^{2} x \left(x^{2} + 1\right)}{4}}{\frac{d}{d x} \frac{1}{\log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}^{3}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{\left(x^{2} + 1\right) \left(\frac{\pi^{2} x^{2}}{2} + \frac{\pi^{2} \left(x^{2} + 1\right)}{4}\right) \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}^{4} \operatorname{atan}{\left(x \right)}}{3}\right)$$
=
$$\lim_{x \to \infty}\left(\left(\frac{3 \pi^{2} x^{2}}{4} + \frac{\pi^{2}}{4}\right) \left(- \frac{\pi x^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{4}}{6} - \frac{2 \pi x^{2} \log{\left(2 \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{3}}{3} + \frac{2 \pi x^{2} \log{\left(\pi \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{3}}{3} - \pi x^{2} \log{\left(\pi \right)}^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{2} - \pi x^{2} \log{\left(2 \right)}^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{2} + 2 \pi x^{2} \log{\left(2 \right)} \log{\left(\pi \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{2} - 2 \pi x^{2} \log{\left(2 \right)} \log{\left(\pi \right)}^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)} - \frac{2 \pi x^{2} \log{\left(2 \right)}^{3} \log{\left(\operatorname{atan}{\left(x \right)} \right)}}{3} + \frac{2 \pi x^{2} \log{\left(\pi \right)}^{3} \log{\left(\operatorname{atan}{\left(x \right)} \right)}}{3} + 2 \pi x^{2} \log{\left(2 \right)}^{2} \log{\left(\pi \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)} - \pi x^{2} \log{\left(2 \right)}^{2} \log{\left(\pi \right)}^{2} - \frac{\pi x^{2} \log{\left(\pi \right)}^{4}}{6} - \frac{\pi x^{2} \log{\left(2 \right)}^{4}}{6} + \frac{2 \pi x^{2} \log{\left(2 \right)}^{3} \log{\left(\pi \right)}}{3} + \frac{2 \pi x^{2} \log{\left(2 \right)} \log{\left(\pi \right)}^{3}}{3} - \frac{\pi \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{4}}{6} - \frac{2 \pi \log{\left(2 \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{3}}{3} + \frac{2 \pi \log{\left(\pi \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{3}}{3} - \pi \log{\left(\pi \right)}^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{2} - \pi \log{\left(2 \right)}^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{2} + 2 \pi \log{\left(2 \right)} \log{\left(\pi \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{2} - 2 \pi \log{\left(2 \right)} \log{\left(\pi \right)}^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)} - \frac{2 \pi \log{\left(2 \right)}^{3} \log{\left(\operatorname{atan}{\left(x \right)} \right)}}{3} + \frac{2 \pi \log{\left(\pi \right)}^{3} \log{\left(\operatorname{atan}{\left(x \right)} \right)}}{3} + 2 \pi \log{\left(2 \right)}^{2} \log{\left(\pi \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)} - \pi \log{\left(2 \right)}^{2} \log{\left(\pi \right)}^{2} - \frac{\pi \log{\left(\pi \right)}^{4}}{6} - \frac{\pi \log{\left(2 \right)}^{4}}{6} + \frac{2 \pi \log{\left(2 \right)}^{3} \log{\left(\pi \right)}}{3} + \frac{2 \pi \log{\left(2 \right)} \log{\left(\pi \right)}^{3}}{3}\right)\right)$$
=
$$\lim_{x \to \infty}\left(\left(\frac{3 \pi^{2} x^{2}}{4} + \frac{\pi^{2}}{4}\right) \left(- \frac{\pi x^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{4}}{6} - \frac{2 \pi x^{2} \log{\left(2 \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{3}}{3} + \frac{2 \pi x^{2} \log{\left(\pi \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{3}}{3} - \pi x^{2} \log{\left(\pi \right)}^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{2} - \pi x^{2} \log{\left(2 \right)}^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{2} + 2 \pi x^{2} \log{\left(2 \right)} \log{\left(\pi \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{2} - 2 \pi x^{2} \log{\left(2 \right)} \log{\left(\pi \right)}^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)} - \frac{2 \pi x^{2} \log{\left(2 \right)}^{3} \log{\left(\operatorname{atan}{\left(x \right)} \right)}}{3} + \frac{2 \pi x^{2} \log{\left(\pi \right)}^{3} \log{\left(\operatorname{atan}{\left(x \right)} \right)}}{3} + 2 \pi x^{2} \log{\left(2 \right)}^{2} \log{\left(\pi \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)} - \pi x^{2} \log{\left(2 \right)}^{2} \log{\left(\pi \right)}^{2} - \frac{\pi x^{2} \log{\left(\pi \right)}^{4}}{6} - \frac{\pi x^{2} \log{\left(2 \right)}^{4}}{6} + \frac{2 \pi x^{2} \log{\left(2 \right)}^{3} \log{\left(\pi \right)}}{3} + \frac{2 \pi x^{2} \log{\left(2 \right)} \log{\left(\pi \right)}^{3}}{3} - \frac{\pi \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{4}}{6} - \frac{2 \pi \log{\left(2 \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{3}}{3} + \frac{2 \pi \log{\left(\pi \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{3}}{3} - \pi \log{\left(\pi \right)}^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{2} - \pi \log{\left(2 \right)}^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{2} + 2 \pi \log{\left(2 \right)} \log{\left(\pi \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{2} - 2 \pi \log{\left(2 \right)} \log{\left(\pi \right)}^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)} - \frac{2 \pi \log{\left(2 \right)}^{3} \log{\left(\operatorname{atan}{\left(x \right)} \right)}}{3} + \frac{2 \pi \log{\left(\pi \right)}^{3} \log{\left(\operatorname{atan}{\left(x \right)} \right)}}{3} + 2 \pi \log{\left(2 \right)}^{2} \log{\left(\pi \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)} - \pi \log{\left(2 \right)}^{2} \log{\left(\pi \right)}^{2} - \frac{\pi \log{\left(\pi \right)}^{4}}{6} - \frac{\pi \log{\left(2 \right)}^{4}}{6} + \frac{2 \pi \log{\left(2 \right)}^{3} \log{\left(\pi \right)}}{3} + \frac{2 \pi \log{\left(2 \right)} \log{\left(\pi \right)}^{3}}{3}\right)\right)$$
=
$$- \frac{2}{\pi}$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)
oo/-oo,
tal que el límite para el numerador es
$$\lim_{x \to \infty} x = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} \frac{1}{\log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}} = -\infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(x \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \infty}\left(x \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \frac{1}{\log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \left(x^{2} + 1\right) \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}^{2} \operatorname{atan}{\left(x \right)}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{\pi \left(x^{2} + 1\right) \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}^{2}}{2}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- \frac{\pi \left(x^{2} + 1\right)}{2}\right)}{\frac{d}{d x} \frac{1}{\log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}^{2}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\pi x \left(x^{2} + 1\right) \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}^{3} \operatorname{atan}{\left(x \right)}}{2}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\pi^{2} x \left(x^{2} + 1\right) \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}^{3}}{4}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{\pi^{2} x \left(x^{2} + 1\right)}{4}}{\frac{d}{d x} \frac{1}{\log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}^{3}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{\left(x^{2} + 1\right) \left(\frac{\pi^{2} x^{2}}{2} + \frac{\pi^{2} \left(x^{2} + 1\right)}{4}\right) \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}^{4} \operatorname{atan}{\left(x \right)}}{3}\right)$$
=
$$\lim_{x \to \infty}\left(\left(\frac{3 \pi^{2} x^{2}}{4} + \frac{\pi^{2}}{4}\right) \left(- \frac{\pi x^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{4}}{6} - \frac{2 \pi x^{2} \log{\left(2 \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{3}}{3} + \frac{2 \pi x^{2} \log{\left(\pi \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{3}}{3} - \pi x^{2} \log{\left(\pi \right)}^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{2} - \pi x^{2} \log{\left(2 \right)}^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{2} + 2 \pi x^{2} \log{\left(2 \right)} \log{\left(\pi \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{2} - 2 \pi x^{2} \log{\left(2 \right)} \log{\left(\pi \right)}^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)} - \frac{2 \pi x^{2} \log{\left(2 \right)}^{3} \log{\left(\operatorname{atan}{\left(x \right)} \right)}}{3} + \frac{2 \pi x^{2} \log{\left(\pi \right)}^{3} \log{\left(\operatorname{atan}{\left(x \right)} \right)}}{3} + 2 \pi x^{2} \log{\left(2 \right)}^{2} \log{\left(\pi \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)} - \pi x^{2} \log{\left(2 \right)}^{2} \log{\left(\pi \right)}^{2} - \frac{\pi x^{2} \log{\left(\pi \right)}^{4}}{6} - \frac{\pi x^{2} \log{\left(2 \right)}^{4}}{6} + \frac{2 \pi x^{2} \log{\left(2 \right)}^{3} \log{\left(\pi \right)}}{3} + \frac{2 \pi x^{2} \log{\left(2 \right)} \log{\left(\pi \right)}^{3}}{3} - \frac{\pi \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{4}}{6} - \frac{2 \pi \log{\left(2 \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{3}}{3} + \frac{2 \pi \log{\left(\pi \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{3}}{3} - \pi \log{\left(\pi \right)}^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{2} - \pi \log{\left(2 \right)}^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{2} + 2 \pi \log{\left(2 \right)} \log{\left(\pi \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{2} - 2 \pi \log{\left(2 \right)} \log{\left(\pi \right)}^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)} - \frac{2 \pi \log{\left(2 \right)}^{3} \log{\left(\operatorname{atan}{\left(x \right)} \right)}}{3} + \frac{2 \pi \log{\left(\pi \right)}^{3} \log{\left(\operatorname{atan}{\left(x \right)} \right)}}{3} + 2 \pi \log{\left(2 \right)}^{2} \log{\left(\pi \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)} - \pi \log{\left(2 \right)}^{2} \log{\left(\pi \right)}^{2} - \frac{\pi \log{\left(\pi \right)}^{4}}{6} - \frac{\pi \log{\left(2 \right)}^{4}}{6} + \frac{2 \pi \log{\left(2 \right)}^{3} \log{\left(\pi \right)}}{3} + \frac{2 \pi \log{\left(2 \right)} \log{\left(\pi \right)}^{3}}{3}\right)\right)$$
=
$$\lim_{x \to \infty}\left(\left(\frac{3 \pi^{2} x^{2}}{4} + \frac{\pi^{2}}{4}\right) \left(- \frac{\pi x^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{4}}{6} - \frac{2 \pi x^{2} \log{\left(2 \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{3}}{3} + \frac{2 \pi x^{2} \log{\left(\pi \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{3}}{3} - \pi x^{2} \log{\left(\pi \right)}^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{2} - \pi x^{2} \log{\left(2 \right)}^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{2} + 2 \pi x^{2} \log{\left(2 \right)} \log{\left(\pi \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{2} - 2 \pi x^{2} \log{\left(2 \right)} \log{\left(\pi \right)}^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)} - \frac{2 \pi x^{2} \log{\left(2 \right)}^{3} \log{\left(\operatorname{atan}{\left(x \right)} \right)}}{3} + \frac{2 \pi x^{2} \log{\left(\pi \right)}^{3} \log{\left(\operatorname{atan}{\left(x \right)} \right)}}{3} + 2 \pi x^{2} \log{\left(2 \right)}^{2} \log{\left(\pi \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)} - \pi x^{2} \log{\left(2 \right)}^{2} \log{\left(\pi \right)}^{2} - \frac{\pi x^{2} \log{\left(\pi \right)}^{4}}{6} - \frac{\pi x^{2} \log{\left(2 \right)}^{4}}{6} + \frac{2 \pi x^{2} \log{\left(2 \right)}^{3} \log{\left(\pi \right)}}{3} + \frac{2 \pi x^{2} \log{\left(2 \right)} \log{\left(\pi \right)}^{3}}{3} - \frac{\pi \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{4}}{6} - \frac{2 \pi \log{\left(2 \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{3}}{3} + \frac{2 \pi \log{\left(\pi \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{3}}{3} - \pi \log{\left(\pi \right)}^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{2} - \pi \log{\left(2 \right)}^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{2} + 2 \pi \log{\left(2 \right)} \log{\left(\pi \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)}^{2} - 2 \pi \log{\left(2 \right)} \log{\left(\pi \right)}^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)} - \frac{2 \pi \log{\left(2 \right)}^{3} \log{\left(\operatorname{atan}{\left(x \right)} \right)}}{3} + \frac{2 \pi \log{\left(\pi \right)}^{3} \log{\left(\operatorname{atan}{\left(x \right)} \right)}}{3} + 2 \pi \log{\left(2 \right)}^{2} \log{\left(\pi \right)} \log{\left(\operatorname{atan}{\left(x \right)} \right)} - \pi \log{\left(2 \right)}^{2} \log{\left(\pi \right)}^{2} - \frac{\pi \log{\left(\pi \right)}^{4}}{6} - \frac{\pi \log{\left(2 \right)}^{4}}{6} + \frac{2 \pi \log{\left(2 \right)}^{3} \log{\left(\pi \right)}}{3} + \frac{2 \pi \log{\left(2 \right)} \log{\left(\pi \right)}^{3}}{3}\right)\right)$$
=
$$- \frac{2}{\pi}$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)
Gráfica
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(x \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}\right) = - \frac{2}{\pi}$$
$$\lim_{x \to 0^-}\left(x \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}\right) = 0$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(x \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}\right) = 0$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(x \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}\right) = - \log{\left(2 \right)}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(x \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}\right) = - \log{\left(2 \right)}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(x \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}\right) = - \infty i$$
Más detalles con x→-oo
$$\lim_{x \to 0^-}\left(x \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}\right) = 0$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(x \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}\right) = 0$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(x \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}\right) = - \log{\left(2 \right)}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(x \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}\right) = - \log{\left(2 \right)}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(x \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}\right) = - \infty i$$
Más detalles con x→-oo
Gráfico