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Otras calculadoras:
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- Límite de la función:
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Límite de ((-7+2*x^2+21*x)/(9+2*x^2+18*x))^(1+2*x)
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Límite de ((2+2*x^2)/(1+2*x^2))^(x^2)
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Límite de (2-cos(3*x))^(1/log(1+x^2))
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Límite de (2-4*x)/(sqrt(x)-sqrt(2)/2)
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Expresiones idénticas
- atan(pi/(tres + tres *x))^(uno +x)*atan(pi/(tres *x))^(-x)
- arco tangente de gente de ( número pi dividir por (3 más 3 multiplicar por x)) en el grado (1 más x) multiplicar por arco tangente de gente de ( número pi dividir por (3 multiplicar por x)) en el grado ( menos x)
- arco tangente de gente de ( número pi dividir por (tres más tres multiplicar por x)) en el grado (uno más x) multiplicar por arco tangente de gente de ( número pi dividir por (tres multiplicar por x)) en el grado ( menos x)
- atan(pi/(3+3*x))(1+x)*atan(pi/(3*x))(-x)
- atanpi/3+3*x1+x*atanpi/3*x-x
- atan(pi/(3+3x))^(1+x)atan(pi/(3x))^(-x)
- atan(pi/(3+3x))(1+x)atan(pi/(3x))(-x)
- atanpi/3+3x1+xatanpi/3x-x
- atanpi/3+3x^1+xatanpi/3x^-x
- atan(pi dividir por (3+3*x))^(1+x)*atan(pi dividir por (3*x))^(-x)
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Expresiones semejantes
- atan(pi/(3+3*x))^(1+x)*atan(pi/(3*x))^(x)
- atan(pi/(3+3*x))^(1-x)*atan(pi/(3*x))^(-x)
- atan(pi/(3-3*x))^(1+x)*atan(pi/(3*x))^(-x)
- arctan(pi/(3+3*x))^(1+x)*arctan(pi/(3*x))^(-x)
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Expresiones con funciones
- Arcotangente arctan
- atan(sqrt(x))/sqrt(x)
- atan(n/(1+n))^(n+1/n)*atan((1+n)/(2+n))^(-1-n-1/(1+n))
- atan(x)/(1+x^(-7))^(1/8)
- atan(10*x)/tan(10*x)
- atan(2*x)*sin(2*x)
- Número Pi pi
- Piecewise(((-3+x)^2,x>3),(5,x=3),(3-x,x<3),(0,True))
- pi/8
- pi^2/(8*x)
- pi-2*acot(x)*log(x)
- pi/(6+2*x)
- Arcotangente arctan
- atan(sqrt(x))/sqrt(x)
- atan(n/(1+n))^(n+1/n)*atan((1+n)/(2+n))^(-1-n-1/(1+n))
- atan(x)/(1+x^(-7))^(1/8)
- atan(10*x)/tan(10*x)
- atan(2*x)*sin(2*x)
- Número Pi pi
- Piecewise(((-3+x)^2,x>3),(5,x=3),(3-x,x<3),(0,True))
- pi/8
- pi^2/(8*x)
- pi-2*acot(x)*log(x)
- pi/(6+2*x)
Límite de la función atan(pi/(3+3*x))^(1+x)*atan(pi/(3*x))^(-x)
Solución
Ha introducido
[src]
/ 1 + x/ pi \ -x/ pi\\ lim |atan |-------|*atan |---|| x->oo\ \3 + 3*x/ \3*x//
$$\lim_{x \to \infty}\left(\operatorname{atan}^{- x}{\left(\frac{\pi}{3 x} \right)} \operatorname{atan}^{x + 1}{\left(\frac{\pi}{3 x + 3} \right)}\right)$$
Limit(atan(pi/(3 + 3*x))^(1 + x)*atan(pi/((3*x)))^(-x), x, oo, dir='-')
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to \infty} \operatorname{atan}^{- x}{\left(\frac{\pi}{3 x} \right)} = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} \operatorname{atan}^{- x - 1}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(\operatorname{atan}^{- x}{\left(\frac{\pi}{3 x} \right)} \operatorname{atan}^{x + 1}{\left(\frac{\pi}{3 x + 3} \right)}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \infty}\left(\operatorname{atan}^{- x}{\left(\frac{\pi}{3 x} \right)} \operatorname{atan}^{x + 1}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \operatorname{atan}^{- x}{\left(\frac{\pi}{3 x} \right)}}{\frac{d}{d x} \operatorname{atan}^{- x - 1}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\left(- \log{\left(\operatorname{atan}{\left(\frac{\pi}{3 x} \right)} \right)} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} \operatorname{atan}^{x}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{\pi \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} \operatorname{atan}^{x}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 x \operatorname{atan}{\left(\frac{\pi}{3 x} \right)} + \frac{\pi^{2} \operatorname{atan}{\left(\frac{\pi}{3 x} \right)}}{3 x}}\right) \operatorname{atan}^{- x}{\left(\frac{\pi}{3 x} \right)}}{\frac{\pi x}{3 x^{2} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{\pi^{2} x^{2} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 \left(x^{2} + 2 x + 1\right)} + 6 x \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{2 \pi^{2} x \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 \left(x^{2} + 2 x + 1\right)} + 3 \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{\pi^{2} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 \left(x^{2} + 2 x + 1\right)}} - \log{\left(\operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} \right)} + \frac{\pi}{3 x^{2} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{\pi^{2} x^{2} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 \left(x^{2} + 2 x + 1\right)} + 6 x \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{2 \pi^{2} x \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 \left(x^{2} + 2 x + 1\right)} + 3 \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{\pi^{2} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 \left(x^{2} + 2 x + 1\right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\left(- \log{\left(\operatorname{atan}{\left(\frac{\pi}{3 x} \right)} \right)} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} \operatorname{atan}^{x}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{\pi \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} \operatorname{atan}^{x}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 x \operatorname{atan}{\left(\frac{\pi}{3 x} \right)} + \frac{\pi^{2} \operatorname{atan}{\left(\frac{\pi}{3 x} \right)}}{3 x}}\right) \operatorname{atan}^{- x}{\left(\frac{\pi}{3 x} \right)}}{\frac{\pi x}{3 x^{2} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{\pi^{2} x^{2} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 \left(x^{2} + 2 x + 1\right)} + 6 x \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{2 \pi^{2} x \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 \left(x^{2} + 2 x + 1\right)} + 3 \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{\pi^{2} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 \left(x^{2} + 2 x + 1\right)}} - \log{\left(\operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} \right)} + \frac{\pi}{3 x^{2} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{\pi^{2} x^{2} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 \left(x^{2} + 2 x + 1\right)} + 6 x \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{2 \pi^{2} x \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 \left(x^{2} + 2 x + 1\right)} + 3 \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{\pi^{2} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 \left(x^{2} + 2 x + 1\right)}}}\right)$$
=
$$0$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 1 vez (veces)
oo/oo,
tal que el límite para el numerador es
$$\lim_{x \to \infty} \operatorname{atan}^{- x}{\left(\frac{\pi}{3 x} \right)} = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} \operatorname{atan}^{- x - 1}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(\operatorname{atan}^{- x}{\left(\frac{\pi}{3 x} \right)} \operatorname{atan}^{x + 1}{\left(\frac{\pi}{3 x + 3} \right)}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \infty}\left(\operatorname{atan}^{- x}{\left(\frac{\pi}{3 x} \right)} \operatorname{atan}^{x + 1}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \operatorname{atan}^{- x}{\left(\frac{\pi}{3 x} \right)}}{\frac{d}{d x} \operatorname{atan}^{- x - 1}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\left(- \log{\left(\operatorname{atan}{\left(\frac{\pi}{3 x} \right)} \right)} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} \operatorname{atan}^{x}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{\pi \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} \operatorname{atan}^{x}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 x \operatorname{atan}{\left(\frac{\pi}{3 x} \right)} + \frac{\pi^{2} \operatorname{atan}{\left(\frac{\pi}{3 x} \right)}}{3 x}}\right) \operatorname{atan}^{- x}{\left(\frac{\pi}{3 x} \right)}}{\frac{\pi x}{3 x^{2} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{\pi^{2} x^{2} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 \left(x^{2} + 2 x + 1\right)} + 6 x \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{2 \pi^{2} x \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 \left(x^{2} + 2 x + 1\right)} + 3 \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{\pi^{2} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 \left(x^{2} + 2 x + 1\right)}} - \log{\left(\operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} \right)} + \frac{\pi}{3 x^{2} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{\pi^{2} x^{2} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 \left(x^{2} + 2 x + 1\right)} + 6 x \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{2 \pi^{2} x \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 \left(x^{2} + 2 x + 1\right)} + 3 \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{\pi^{2} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 \left(x^{2} + 2 x + 1\right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\left(- \log{\left(\operatorname{atan}{\left(\frac{\pi}{3 x} \right)} \right)} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} \operatorname{atan}^{x}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{\pi \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} \operatorname{atan}^{x}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 x \operatorname{atan}{\left(\frac{\pi}{3 x} \right)} + \frac{\pi^{2} \operatorname{atan}{\left(\frac{\pi}{3 x} \right)}}{3 x}}\right) \operatorname{atan}^{- x}{\left(\frac{\pi}{3 x} \right)}}{\frac{\pi x}{3 x^{2} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{\pi^{2} x^{2} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 \left(x^{2} + 2 x + 1\right)} + 6 x \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{2 \pi^{2} x \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 \left(x^{2} + 2 x + 1\right)} + 3 \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{\pi^{2} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 \left(x^{2} + 2 x + 1\right)}} - \log{\left(\operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} \right)} + \frac{\pi}{3 x^{2} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{\pi^{2} x^{2} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 \left(x^{2} + 2 x + 1\right)} + 6 x \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{2 \pi^{2} x \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 \left(x^{2} + 2 x + 1\right)} + 3 \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)} + \frac{\pi^{2} \operatorname{atan}{\left(\frac{\pi}{3 \left(x + 1\right)} \right)}}{3 \left(x^{2} + 2 x + 1\right)}}}\right)$$
=
$$0$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 1 vez (veces)
Gráfica
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(\operatorname{atan}^{- x}{\left(\frac{\pi}{3 x} \right)} \operatorname{atan}^{x + 1}{\left(\frac{\pi}{3 x + 3} \right)}\right) = 0$$
$$\lim_{x \to 0^-}\left(\operatorname{atan}^{- x}{\left(\frac{\pi}{3 x} \right)} \operatorname{atan}^{x + 1}{\left(\frac{\pi}{3 x + 3} \right)}\right) = \operatorname{atan}{\left(\frac{\pi}{3} \right)}$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\operatorname{atan}^{- x}{\left(\frac{\pi}{3 x} \right)} \operatorname{atan}^{x + 1}{\left(\frac{\pi}{3 x + 3} \right)}\right) = \operatorname{atan}{\left(\frac{\pi}{3} \right)}$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\operatorname{atan}^{- x}{\left(\frac{\pi}{3 x} \right)} \operatorname{atan}^{x + 1}{\left(\frac{\pi}{3 x + 3} \right)}\right) = \frac{\operatorname{atan}^{2}{\left(\frac{\pi}{6} \right)}}{\operatorname{atan}{\left(\frac{\pi}{3} \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\operatorname{atan}^{- x}{\left(\frac{\pi}{3 x} \right)} \operatorname{atan}^{x + 1}{\left(\frac{\pi}{3 x + 3} \right)}\right) = \frac{\operatorname{atan}^{2}{\left(\frac{\pi}{6} \right)}}{\operatorname{atan}{\left(\frac{\pi}{3} \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\operatorname{atan}^{- x}{\left(\frac{\pi}{3 x} \right)} \operatorname{atan}^{x + 1}{\left(\frac{\pi}{3 x + 3} \right)}\right) = 0$$
Más detalles con x→-oo
$$\lim_{x \to 0^-}\left(\operatorname{atan}^{- x}{\left(\frac{\pi}{3 x} \right)} \operatorname{atan}^{x + 1}{\left(\frac{\pi}{3 x + 3} \right)}\right) = \operatorname{atan}{\left(\frac{\pi}{3} \right)}$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\operatorname{atan}^{- x}{\left(\frac{\pi}{3 x} \right)} \operatorname{atan}^{x + 1}{\left(\frac{\pi}{3 x + 3} \right)}\right) = \operatorname{atan}{\left(\frac{\pi}{3} \right)}$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\operatorname{atan}^{- x}{\left(\frac{\pi}{3 x} \right)} \operatorname{atan}^{x + 1}{\left(\frac{\pi}{3 x + 3} \right)}\right) = \frac{\operatorname{atan}^{2}{\left(\frac{\pi}{6} \right)}}{\operatorname{atan}{\left(\frac{\pi}{3} \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\operatorname{atan}^{- x}{\left(\frac{\pi}{3 x} \right)} \operatorname{atan}^{x + 1}{\left(\frac{\pi}{3 x + 3} \right)}\right) = \frac{\operatorname{atan}^{2}{\left(\frac{\pi}{6} \right)}}{\operatorname{atan}{\left(\frac{\pi}{3} \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\operatorname{atan}^{- x}{\left(\frac{\pi}{3 x} \right)} \operatorname{atan}^{x + 1}{\left(\frac{\pi}{3 x + 3} \right)}\right) = 0$$
Más detalles con x→-oo