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- Límite de la función:
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Límite de x/(-2+x)
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Límite de x^(-2)
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Límite de (-4+x^2)/(6+x^2-5*x)
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Límite de (2-sqrt(-3+x))/(-49+x^2)
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Expresiones idénticas
- log(pi/ dos -x)/tan(x)
- logaritmo de ( número pi dividir por 2 menos x) dividir por tangente de (x)
- logaritmo de ( número pi dividir por dos menos x) dividir por tangente de (x)
- logpi/2-x/tanx
- log(pi dividir por 2-x) dividir por tan(x)
-
Expresiones semejantes
- log(pi/2+x)/tan(x)
-
Expresiones con funciones
- Logaritmo log
- log(1+x)/x
- log(x)^(1/x)
- log(-1+x)
- log(1-7*x)/sin(pi*(7+x))
- log(1+x^2)
- Número Pi pi
- pi/(x*cot(pi*x/2))
- pi*n*x*sin(-3/2+x/2)/(6*cot(a))
- pi/atan(x)
- Piecewise((6-3*x,x<2),(6-2*x^2,True))
- pi^2*x^2*(-2+x)^2/2
- Tangente tan
- tan(x)/tan(3*x)
- tan(3*x)/(2*sin(x))
- tan(5*x)/sin(2*x)
- tan(4*x)/sin(x)
- tan(k*x)/x
Límite de la función log(pi/2-x)/tan(x)
Solución
Ha introducido
[src]
/ /pi \\
|log|-- - x||
| \2 /|
lim |-----------|
pi \ tan(x) /
x->--+
2
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\log{\left(- x + \frac{\pi}{2} \right)}}{\tan{\left(x \right)}}\right)$$
Limit(log(pi/2 - x)/tan(x), x, pi/2)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to \frac{\pi}{2}^+} \frac{1}{\tan{\left(x \right)}} = 0$$
y el límite para el denominador es
$$\lim_{x \to \frac{\pi}{2}^+} \frac{1}{\log{\left(\frac{\pi - 2 x}{2} \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\log{\left(- x + \frac{\pi}{2} \right)}}{\tan{\left(x \right)}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\log{\left(\frac{\pi - 2 x}{2} \right)}}{\tan{\left(x \right)}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{1}{\tan{\left(x \right)}}}{\frac{d}{d x} \frac{1}{\log{\left(\frac{\pi - 2 x}{2} \right)}}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\left(\pi - 2 x\right) \left(- \tan^{2}{\left(x \right)} - 1\right) \log{\left(- x + \frac{\pi}{2} \right)}^{2}}{2 \tan^{2}{\left(x \right)}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{\left(\pi - 2 x\right) \left(- \tan^{2}{\left(x \right)} - 1\right)}{2 \tan^{2}{\left(x \right)}}}{\frac{d}{d x} \frac{1}{\log{\left(- x + \frac{\pi}{2} \right)}^{2}}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\left(- x + \frac{\pi}{2}\right) \left(\frac{\left(\pi - 2 x\right) \left(- 2 \tan^{2}{\left(x \right)} - 2\right) \left(- \tan^{2}{\left(x \right)} - 1\right)}{2 \tan^{3}{\left(x \right)}} - \frac{\left(\pi - 2 x\right) \left(2 \tan^{2}{\left(x \right)} + 2\right)}{2 \tan{\left(x \right)}} - \frac{- \tan^{2}{\left(x \right)} - 1}{\tan^{2}{\left(x \right)}}\right) \log{\left(- x + \frac{\pi}{2} \right)}^{3}}{2}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{\left(- x + \frac{\pi}{2}\right) \log{\left(- x + \frac{\pi}{2} \right)}^{3}}{2}}{\frac{d}{d x} \frac{1}{\frac{\left(\pi - 2 x\right) \left(- 2 \tan^{2}{\left(x \right)} - 2\right) \left(- \tan^{2}{\left(x \right)} - 1\right)}{2 \tan^{3}{\left(x \right)}} - \frac{\left(\pi - 2 x\right) \left(2 \tan^{2}{\left(x \right)} + 2\right)}{2 \tan{\left(x \right)}} - \frac{- \tan^{2}{\left(x \right)} - 1}{\tan^{2}{\left(x \right)}}}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\left(- \frac{\log{\left(- x + \frac{\pi}{2} \right)}^{3}}{2} - \frac{3 \log{\left(- x + \frac{\pi}{2} \right)}^{2}}{2}\right) \left(\frac{\left(\pi - 2 x\right) \left(- 2 \tan^{2}{\left(x \right)} - 2\right) \left(- \tan^{2}{\left(x \right)} - 1\right)}{2 \tan^{3}{\left(x \right)}} - \frac{\left(\pi - 2 x\right) \left(2 \tan^{2}{\left(x \right)} + 2\right)}{2 \tan{\left(x \right)}} - \frac{- \tan^{2}{\left(x \right)} - 1}{\tan^{2}{\left(x \right)}}\right)^{2}}{- \frac{\left(\pi - 2 x\right) \left(- 3 \tan^{2}{\left(x \right)} - 3\right) \left(- 2 \tan^{2}{\left(x \right)} - 2\right) \left(- \tan^{2}{\left(x \right)} - 1\right)}{2 \tan^{4}{\left(x \right)}} + \frac{\left(\pi - 2 x\right) \left(- 2 \tan^{2}{\left(x \right)} - 2\right) \left(2 \tan^{2}{\left(x \right)} + 2\right)}{2 \tan^{2}{\left(x \right)}} - \left(\pi - 2 x\right) \left(- 2 \tan^{2}{\left(x \right)} - 2\right) + \frac{3 \left(\pi - 2 x\right) \left(- \tan^{2}{\left(x \right)} - 1\right) \left(2 \tan^{2}{\left(x \right)} + 2\right)}{2 \tan^{2}{\left(x \right)}} + \frac{2 \left(- 2 \tan^{2}{\left(x \right)} - 2\right) \left(- \tan^{2}{\left(x \right)} - 1\right)}{\tan^{3}{\left(x \right)}} - \frac{2 \left(2 \tan^{2}{\left(x \right)} + 2\right)}{\tan{\left(x \right)}}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{- \frac{\log{\left(- x + \frac{\pi}{2} \right)}^{3}}{2} - \frac{3 \log{\left(- x + \frac{\pi}{2} \right)}^{2}}{2}}{- 2 x - \frac{8 x}{\tan^{2}{\left(x \right)}} - \frac{6 x}{\tan^{4}{\left(x \right)}} + \pi + \frac{4}{\tan{\left(x \right)}} + \frac{4 \pi}{\tan^{2}{\left(x \right)}} + \frac{4}{\tan^{3}{\left(x \right)}} + \frac{3 \pi}{\tan^{4}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{- \frac{\log{\left(- x + \frac{\pi}{2} \right)}^{3}}{2} - \frac{3 \log{\left(- x + \frac{\pi}{2} \right)}^{2}}{2}}{- 2 x - \frac{8 x}{\tan^{2}{\left(x \right)}} - \frac{6 x}{\tan^{4}{\left(x \right)}} + \pi + \frac{4}{\tan{\left(x \right)}} + \frac{4 \pi}{\tan^{2}{\left(x \right)}} + \frac{4}{\tan^{3}{\left(x \right)}} + \frac{3 \pi}{\tan^{4}{\left(x \right)}}}\right)$$
=
$$0$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)
0/0,
tal que el límite para el numerador es
$$\lim_{x \to \frac{\pi}{2}^+} \frac{1}{\tan{\left(x \right)}} = 0$$
y el límite para el denominador es
$$\lim_{x \to \frac{\pi}{2}^+} \frac{1}{\log{\left(\frac{\pi - 2 x}{2} \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\log{\left(- x + \frac{\pi}{2} \right)}}{\tan{\left(x \right)}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\log{\left(\frac{\pi - 2 x}{2} \right)}}{\tan{\left(x \right)}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{1}{\tan{\left(x \right)}}}{\frac{d}{d x} \frac{1}{\log{\left(\frac{\pi - 2 x}{2} \right)}}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\left(\pi - 2 x\right) \left(- \tan^{2}{\left(x \right)} - 1\right) \log{\left(- x + \frac{\pi}{2} \right)}^{2}}{2 \tan^{2}{\left(x \right)}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{\left(\pi - 2 x\right) \left(- \tan^{2}{\left(x \right)} - 1\right)}{2 \tan^{2}{\left(x \right)}}}{\frac{d}{d x} \frac{1}{\log{\left(- x + \frac{\pi}{2} \right)}^{2}}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\left(- x + \frac{\pi}{2}\right) \left(\frac{\left(\pi - 2 x\right) \left(- 2 \tan^{2}{\left(x \right)} - 2\right) \left(- \tan^{2}{\left(x \right)} - 1\right)}{2 \tan^{3}{\left(x \right)}} - \frac{\left(\pi - 2 x\right) \left(2 \tan^{2}{\left(x \right)} + 2\right)}{2 \tan{\left(x \right)}} - \frac{- \tan^{2}{\left(x \right)} - 1}{\tan^{2}{\left(x \right)}}\right) \log{\left(- x + \frac{\pi}{2} \right)}^{3}}{2}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{\left(- x + \frac{\pi}{2}\right) \log{\left(- x + \frac{\pi}{2} \right)}^{3}}{2}}{\frac{d}{d x} \frac{1}{\frac{\left(\pi - 2 x\right) \left(- 2 \tan^{2}{\left(x \right)} - 2\right) \left(- \tan^{2}{\left(x \right)} - 1\right)}{2 \tan^{3}{\left(x \right)}} - \frac{\left(\pi - 2 x\right) \left(2 \tan^{2}{\left(x \right)} + 2\right)}{2 \tan{\left(x \right)}} - \frac{- \tan^{2}{\left(x \right)} - 1}{\tan^{2}{\left(x \right)}}}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\left(- \frac{\log{\left(- x + \frac{\pi}{2} \right)}^{3}}{2} - \frac{3 \log{\left(- x + \frac{\pi}{2} \right)}^{2}}{2}\right) \left(\frac{\left(\pi - 2 x\right) \left(- 2 \tan^{2}{\left(x \right)} - 2\right) \left(- \tan^{2}{\left(x \right)} - 1\right)}{2 \tan^{3}{\left(x \right)}} - \frac{\left(\pi - 2 x\right) \left(2 \tan^{2}{\left(x \right)} + 2\right)}{2 \tan{\left(x \right)}} - \frac{- \tan^{2}{\left(x \right)} - 1}{\tan^{2}{\left(x \right)}}\right)^{2}}{- \frac{\left(\pi - 2 x\right) \left(- 3 \tan^{2}{\left(x \right)} - 3\right) \left(- 2 \tan^{2}{\left(x \right)} - 2\right) \left(- \tan^{2}{\left(x \right)} - 1\right)}{2 \tan^{4}{\left(x \right)}} + \frac{\left(\pi - 2 x\right) \left(- 2 \tan^{2}{\left(x \right)} - 2\right) \left(2 \tan^{2}{\left(x \right)} + 2\right)}{2 \tan^{2}{\left(x \right)}} - \left(\pi - 2 x\right) \left(- 2 \tan^{2}{\left(x \right)} - 2\right) + \frac{3 \left(\pi - 2 x\right) \left(- \tan^{2}{\left(x \right)} - 1\right) \left(2 \tan^{2}{\left(x \right)} + 2\right)}{2 \tan^{2}{\left(x \right)}} + \frac{2 \left(- 2 \tan^{2}{\left(x \right)} - 2\right) \left(- \tan^{2}{\left(x \right)} - 1\right)}{\tan^{3}{\left(x \right)}} - \frac{2 \left(2 \tan^{2}{\left(x \right)} + 2\right)}{\tan{\left(x \right)}}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{- \frac{\log{\left(- x + \frac{\pi}{2} \right)}^{3}}{2} - \frac{3 \log{\left(- x + \frac{\pi}{2} \right)}^{2}}{2}}{- 2 x - \frac{8 x}{\tan^{2}{\left(x \right)}} - \frac{6 x}{\tan^{4}{\left(x \right)}} + \pi + \frac{4}{\tan{\left(x \right)}} + \frac{4 \pi}{\tan^{2}{\left(x \right)}} + \frac{4}{\tan^{3}{\left(x \right)}} + \frac{3 \pi}{\tan^{4}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{- \frac{\log{\left(- x + \frac{\pi}{2} \right)}^{3}}{2} - \frac{3 \log{\left(- x + \frac{\pi}{2} \right)}^{2}}{2}}{- 2 x - \frac{8 x}{\tan^{2}{\left(x \right)}} - \frac{6 x}{\tan^{4}{\left(x \right)}} + \pi + \frac{4}{\tan{\left(x \right)}} + \frac{4 \pi}{\tan^{2}{\left(x \right)}} + \frac{4}{\tan^{3}{\left(x \right)}} + \frac{3 \pi}{\tan^{4}{\left(x \right)}}}\right)$$
=
$$0$$
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
$$\lim_{x \to \frac{\pi}{2}^-}\left(\frac{\log{\left(- x + \frac{\pi}{2} \right)}}{\tan{\left(x \right)}}\right) = 0$$
Más detalles con x→pi/2 a la izquierda
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\log{\left(- x + \frac{\pi}{2} \right)}}{\tan{\left(x \right)}}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(- x + \frac{\pi}{2} \right)}}{\tan{\left(x \right)}}\right)$$
Más detalles con x→oo
$$\lim_{x \to 0^-}\left(\frac{\log{\left(- x + \frac{\pi}{2} \right)}}{\tan{\left(x \right)}}\right) = -\infty$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\log{\left(- x + \frac{\pi}{2} \right)}}{\tan{\left(x \right)}}\right) = \infty$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{\log{\left(- x + \frac{\pi}{2} \right)}}{\tan{\left(x \right)}}\right) = \frac{\log{\left(-1 + \frac{\pi}{2} \right)}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\log{\left(- x + \frac{\pi}{2} \right)}}{\tan{\left(x \right)}}\right) = \frac{\log{\left(-1 + \frac{\pi}{2} \right)}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\log{\left(- x + \frac{\pi}{2} \right)}}{\tan{\left(x \right)}}\right)$$
Más detalles con x→-oo
Más detalles con x→pi/2 a la izquierda
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\log{\left(- x + \frac{\pi}{2} \right)}}{\tan{\left(x \right)}}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(- x + \frac{\pi}{2} \right)}}{\tan{\left(x \right)}}\right)$$
Más detalles con x→oo
$$\lim_{x \to 0^-}\left(\frac{\log{\left(- x + \frac{\pi}{2} \right)}}{\tan{\left(x \right)}}\right) = -\infty$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\log{\left(- x + \frac{\pi}{2} \right)}}{\tan{\left(x \right)}}\right) = \infty$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{\log{\left(- x + \frac{\pi}{2} \right)}}{\tan{\left(x \right)}}\right) = \frac{\log{\left(-1 + \frac{\pi}{2} \right)}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\log{\left(- x + \frac{\pi}{2} \right)}}{\tan{\left(x \right)}}\right) = \frac{\log{\left(-1 + \frac{\pi}{2} \right)}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\log{\left(- x + \frac{\pi}{2} \right)}}{\tan{\left(x \right)}}\right)$$
Más detalles con x→-oo
A la izquierda y a la derecha
[src]
/ /pi \\
|log|-- - x||
| \2 /|
lim |-----------|
pi \ tan(x) /
x->--+
2
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\log{\left(- x + \frac{\pi}{2} \right)}}{\tan{\left(x \right)}}\right)$$
0
$$0$$
= (0.00194664737286252 - 0.000844182607821294j)
/ /pi \\
|log|-- - x||
| \2 /|
lim |-----------|
pi \ tan(x) /
x->---
2
$$\lim_{x \to \frac{\pi}{2}^-}\left(\frac{\log{\left(- x + \frac{\pi}{2} \right)}}{\tan{\left(x \right)}}\right)$$
0
$$0$$
= -0.00208709093023149
= -0.00208709093023149
Respuesta numérica
[src]
(0.00194664737286252 - 0.000844182607821294j)
(0.00194664737286252 - 0.000844182607821294j)