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x*log(2*atan(x)/pi)
Límite de la función/ tan(x)/ x*log(2*atan(x)/pi)

Límite de la función x*log(2*atan(x)/pi)

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Solución

Ha introducido [src] 
     /     /2*atan(x)\\
 lim |x*log|---------||
x->oo\     \    pi   //
lim⁡x→∞(xlog⁡(2atan⁡(x)π))\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
oo/-oo,

tal que el límite para el numerador es
lim⁡x→∞x=∞\lim_{x \to \infty} x = \infty
y el límite para el denominador es
lim⁡x→∞1log⁡(2atan⁡(x)π)=−∞\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→∞(xlog⁡(2atan⁡(x)π))\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→∞(xlog⁡(2atan⁡(x)π))\lim_{x \to \infty}\left(x \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}\right)
=
lim⁡x→∞(ddxxddx1log⁡(2atan⁡(x)π))\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→∞(−(x2+1)log⁡(2atan⁡(x)π)2atan⁡(x))\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→∞(−π(x2+1)log⁡(2atan⁡(x)π)22)\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→∞(ddx(−π(x2+1)2)ddx1log⁡(2atan⁡(x)π)2)\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→∞(πx(x2+1)log⁡(2atan⁡(x)π)3atan⁡(x)2)\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→∞(π2x(x2+1)log⁡(2atan⁡(x)π)34)\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→∞(ddxπ2x(x2+1)4ddx1log⁡(2atan⁡(x)π)3)\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→∞(−(x2+1)(π2x22+π2(x2+1)4)log⁡(2atan⁡(x)π)4atan⁡(x)3)\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→∞((3π2x24+π24)(−πx2log⁡(atan⁡(x))46−2πx2log⁡(2)log⁡(atan⁡(x))33+2πx2log⁡(π)log⁡(atan⁡(x))33−πx2log⁡(π)2log⁡(atan⁡(x))2−πx2log⁡(2)2log⁡(atan⁡(x))2+2πx2log⁡(2)log⁡(π)log⁡(atan⁡(x))2−2πx2log⁡(2)log⁡(π)2log⁡(atan⁡(x))−2πx2log⁡(2)3log⁡(atan⁡(x))3+2πx2log⁡(π)3log⁡(atan⁡(x))3+2πx2log⁡(2)2log⁡(π)log⁡(atan⁡(x))−πx2log⁡(2)2log⁡(π)2−πx2log⁡(π)46−πx2log⁡(2)46+2πx2log⁡(2)3log⁡(π)3+2πx2log⁡(2)log⁡(π)33−πlog⁡(atan⁡(x))46−2πlog⁡(2)log⁡(atan⁡(x))33+2πlog⁡(π)log⁡(atan⁡(x))33−πlog⁡(π)2log⁡(atan⁡(x))2−πlog⁡(2)2log⁡(atan⁡(x))2+2πlog⁡(2)log⁡(π)log⁡(atan⁡(x))2−2πlog⁡(2)log⁡(π)2log⁡(atan⁡(x))−2πlog⁡(2)3log⁡(atan⁡(x))3+2πlog⁡(π)3log⁡(atan⁡(x))3+2πlog⁡(2)2log⁡(π)log⁡(atan⁡(x))−πlog⁡(2)2log⁡(π)2−πlog⁡(π)46−πlog⁡(2)46+2πlog⁡(2)3log⁡(π)3+2πlog⁡(2)log⁡(π)33))\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→∞((3π2x24+π24)(−πx2log⁡(atan⁡(x))46−2πx2log⁡(2)log⁡(atan⁡(x))33+2πx2log⁡(π)log⁡(atan⁡(x))33−πx2log⁡(π)2log⁡(atan⁡(x))2−πx2log⁡(2)2log⁡(atan⁡(x))2+2πx2log⁡(2)log⁡(π)log⁡(atan⁡(x))2−2πx2log⁡(2)log⁡(π)2log⁡(atan⁡(x))−2πx2log⁡(2)3log⁡(atan⁡(x))3+2πx2log⁡(π)3log⁡(atan⁡(x))3+2πx2log⁡(2)2log⁡(π)log⁡(atan⁡(x))−πx2log⁡(2)2log⁡(π)2−πx2log⁡(π)46−πx2log⁡(2)46+2πx2log⁡(2)3log⁡(π)3+2πx2log⁡(2)log⁡(π)33−πlog⁡(atan⁡(x))46−2πlog⁡(2)log⁡(atan⁡(x))33+2πlog⁡(π)log⁡(atan⁡(x))33−πlog⁡(π)2log⁡(atan⁡(x))2−πlog⁡(2)2log⁡(atan⁡(x))2+2πlog⁡(2)log⁡(π)log⁡(atan⁡(x))2−2πlog⁡(2)log⁡(π)2log⁡(atan⁡(x))−2πlog⁡(2)3log⁡(atan⁡(x))3+2πlog⁡(π)3log⁡(atan⁡(x))3+2πlog⁡(2)2log⁡(π)log⁡(atan⁡(x))−πlog⁡(2)2log⁡(π)2−πlog⁡(π)46−πlog⁡(2)46+2πlog⁡(2)3log⁡(π)3+2πlog⁡(2)log⁡(π)33))\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)
=
−2π- \frac{2}{\pi}
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)
Gráfica
02468-8-6-4-2-1010-1.00.0
Trazar el gráfico
Otros límites con x→0, -oo, +oo, 1
lim⁡x→∞(xlog⁡(2atan⁡(x)π))=−2π\lim_{x \to \infty}\left(x \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}\right) = - \frac{2}{\pi}
lim⁡x→0−(xlog⁡(2atan⁡(x)π))=0\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→0+(xlog⁡(2atan⁡(x)π))=0\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→1−(xlog⁡(2atan⁡(x)π))=−log⁡(2)\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→1+(xlog⁡(2atan⁡(x)π))=−log⁡(2)\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→−∞(xlog⁡(2atan⁡(x)π))=−∞i\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
Respuesta rápida [src] 
-2 
---
 pi
−2π- \frac{2}{\pi}
Abrir y simplificar
Gráfico
Límite de la función x*log(2*atan(x)/pi)
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