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Expresiones idénticas
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- (1+x)log(log(x))/(xlog(log(1+x)))
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Expresiones con funciones
- Logaritmo log
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- log(-7+2*x)/(-4+x)
- Logaritmo log
- log(sin(x))/log(x)
- log(cos(3*x))/log(cos(4*x))
- log(-cos(x)+sin(x))
- log(-7+4*x)/(-1+3^(-2+x))
- log(-7+2*x)/(-4+x)
- Logaritmo log
- log(sin(x))/log(x)
- log(cos(3*x))/log(cos(4*x))
- log(-cos(x)+sin(x))
- log(-7+4*x)/(-1+3^(-2+x))
- log(-7+2*x)/(-4+x)
- Logaritmo log
- log(sin(x))/log(x)
- log(cos(3*x))/log(cos(4*x))
- log(-cos(x)+sin(x))
- log(-7+4*x)/(-1+3^(-2+x))
- log(-7+2*x)/(-4+x)
Límite de la función (1+x)*log(log(x))/(x*log(log(1+x)))
Solución
Ha introducido
[src]
/(1 + x)*log(log(x))\ lim |-------------------| x->oo\ x*log(log(1 + x)) /
$$\lim_{x \to \infty}\left(\frac{\left(x + 1\right) \log{\left(\log{\left(x \right)} \right)}}{x \log{\left(\log{\left(x + 1 \right)} \right)}}\right)$$
Limit(((1 + x)*log(log(x)))/((x*log(log(1 + x)))), x, oo, dir='-')
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to \infty}\left(\frac{\left(x + 1\right) \log{\left(\log{\left(x \right)} \right)}}{x}\right) = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} \log{\left(\log{\left(x + 1 \right)} \right)} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(\frac{\left(x + 1\right) \log{\left(\log{\left(x \right)} \right)}}{x \log{\left(\log{\left(x + 1 \right)} \right)}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \infty}\left(\frac{\left(x + 1\right) \log{\left(\log{\left(x \right)} \right)}}{x \log{\left(\log{\left(x + 1 \right)} \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{\left(x + 1\right) \log{\left(\log{\left(x \right)} \right)}}{x}}{\frac{d}{d x} \log{\left(\log{\left(x + 1 \right)} \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\left(x + 1\right) \left(\frac{\log{\left(\log{\left(x \right)} \right)}}{x} - \frac{\left(x + 1\right) \log{\left(\log{\left(x \right)} \right)}}{x^{2}} + \frac{x + 1}{x^{2} \log{\left(x \right)}}\right) \log{\left(x + 1 \right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(x + 1\right) \log{\left(x + 1 \right)}}{\frac{d}{d x} \frac{1}{\frac{\log{\left(\log{\left(x \right)} \right)}}{x} - \frac{\left(x + 1\right) \log{\left(\log{\left(x \right)} \right)}}{x^{2}} + \frac{x + 1}{x^{2} \log{\left(x \right)}}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{\log{\left(x + 1 \right)}}{x^{2} \log{\left(x \right)}^{2}} + \frac{1}{x^{2} \log{\left(x \right)}^{2}} - \frac{2 \log{\left(x + 1 \right)} \log{\left(\log{\left(x \right)} \right)}}{x^{3} \log{\left(x \right)}} - \frac{2 \log{\left(\log{\left(x \right)} \right)}}{x^{3} \log{\left(x \right)}} + \frac{2 \log{\left(x + 1 \right)}}{x^{3} \log{\left(x \right)}^{2}} + \frac{2}{x^{3} \log{\left(x \right)}^{2}} + \frac{\log{\left(x + 1 \right)} \log{\left(\log{\left(x \right)} \right)}^{2}}{x^{4}} + \frac{\log{\left(\log{\left(x \right)} \right)}^{2}}{x^{4}} - \frac{2 \log{\left(x + 1 \right)} \log{\left(\log{\left(x \right)} \right)}}{x^{4} \log{\left(x \right)}} - \frac{2 \log{\left(\log{\left(x \right)} \right)}}{x^{4} \log{\left(x \right)}} + \frac{\log{\left(x + 1 \right)}}{x^{4} \log{\left(x \right)}^{2}} + \frac{1}{x^{4} \log{\left(x \right)}^{2}}}{\frac{1}{x^{2} \log{\left(x \right)}} + \frac{1}{x^{2} \log{\left(x \right)}^{2}} - \frac{2 \log{\left(\log{\left(x \right)} \right)}}{x^{3}} + \frac{3}{x^{3} \log{\left(x \right)}} + \frac{1}{x^{3} \log{\left(x \right)}^{2}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{\log{\left(x + 1 \right)}}{x^{2} \log{\left(x \right)}^{2}} + \frac{1}{x^{2} \log{\left(x \right)}^{2}} - \frac{2 \log{\left(x + 1 \right)} \log{\left(\log{\left(x \right)} \right)}}{x^{3} \log{\left(x \right)}} - \frac{2 \log{\left(\log{\left(x \right)} \right)}}{x^{3} \log{\left(x \right)}} + \frac{2 \log{\left(x + 1 \right)}}{x^{3} \log{\left(x \right)}^{2}} + \frac{2}{x^{3} \log{\left(x \right)}^{2}} + \frac{\log{\left(x + 1 \right)} \log{\left(\log{\left(x \right)} \right)}^{2}}{x^{4}} + \frac{\log{\left(\log{\left(x \right)} \right)}^{2}}{x^{4}} - \frac{2 \log{\left(x + 1 \right)} \log{\left(\log{\left(x \right)} \right)}}{x^{4} \log{\left(x \right)}} - \frac{2 \log{\left(\log{\left(x \right)} \right)}}{x^{4} \log{\left(x \right)}} + \frac{\log{\left(x + 1 \right)}}{x^{4} \log{\left(x \right)}^{2}} + \frac{1}{x^{4} \log{\left(x \right)}^{2}}}{\frac{1}{x^{2} \log{\left(x \right)}} + \frac{1}{x^{2} \log{\left(x \right)}^{2}} - \frac{2 \log{\left(\log{\left(x \right)} \right)}}{x^{3}} + \frac{3}{x^{3} \log{\left(x \right)}} + \frac{1}{x^{3} \log{\left(x \right)}^{2}}}\right)$$
=
$$1$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 2 vez (veces)
oo/oo,
tal que el límite para el numerador es
$$\lim_{x \to \infty}\left(\frac{\left(x + 1\right) \log{\left(\log{\left(x \right)} \right)}}{x}\right) = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} \log{\left(\log{\left(x + 1 \right)} \right)} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(\frac{\left(x + 1\right) \log{\left(\log{\left(x \right)} \right)}}{x \log{\left(\log{\left(x + 1 \right)} \right)}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \infty}\left(\frac{\left(x + 1\right) \log{\left(\log{\left(x \right)} \right)}}{x \log{\left(\log{\left(x + 1 \right)} \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{\left(x + 1\right) \log{\left(\log{\left(x \right)} \right)}}{x}}{\frac{d}{d x} \log{\left(\log{\left(x + 1 \right)} \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\left(x + 1\right) \left(\frac{\log{\left(\log{\left(x \right)} \right)}}{x} - \frac{\left(x + 1\right) \log{\left(\log{\left(x \right)} \right)}}{x^{2}} + \frac{x + 1}{x^{2} \log{\left(x \right)}}\right) \log{\left(x + 1 \right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(x + 1\right) \log{\left(x + 1 \right)}}{\frac{d}{d x} \frac{1}{\frac{\log{\left(\log{\left(x \right)} \right)}}{x} - \frac{\left(x + 1\right) \log{\left(\log{\left(x \right)} \right)}}{x^{2}} + \frac{x + 1}{x^{2} \log{\left(x \right)}}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{\log{\left(x + 1 \right)}}{x^{2} \log{\left(x \right)}^{2}} + \frac{1}{x^{2} \log{\left(x \right)}^{2}} - \frac{2 \log{\left(x + 1 \right)} \log{\left(\log{\left(x \right)} \right)}}{x^{3} \log{\left(x \right)}} - \frac{2 \log{\left(\log{\left(x \right)} \right)}}{x^{3} \log{\left(x \right)}} + \frac{2 \log{\left(x + 1 \right)}}{x^{3} \log{\left(x \right)}^{2}} + \frac{2}{x^{3} \log{\left(x \right)}^{2}} + \frac{\log{\left(x + 1 \right)} \log{\left(\log{\left(x \right)} \right)}^{2}}{x^{4}} + \frac{\log{\left(\log{\left(x \right)} \right)}^{2}}{x^{4}} - \frac{2 \log{\left(x + 1 \right)} \log{\left(\log{\left(x \right)} \right)}}{x^{4} \log{\left(x \right)}} - \frac{2 \log{\left(\log{\left(x \right)} \right)}}{x^{4} \log{\left(x \right)}} + \frac{\log{\left(x + 1 \right)}}{x^{4} \log{\left(x \right)}^{2}} + \frac{1}{x^{4} \log{\left(x \right)}^{2}}}{\frac{1}{x^{2} \log{\left(x \right)}} + \frac{1}{x^{2} \log{\left(x \right)}^{2}} - \frac{2 \log{\left(\log{\left(x \right)} \right)}}{x^{3}} + \frac{3}{x^{3} \log{\left(x \right)}} + \frac{1}{x^{3} \log{\left(x \right)}^{2}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{\log{\left(x + 1 \right)}}{x^{2} \log{\left(x \right)}^{2}} + \frac{1}{x^{2} \log{\left(x \right)}^{2}} - \frac{2 \log{\left(x + 1 \right)} \log{\left(\log{\left(x \right)} \right)}}{x^{3} \log{\left(x \right)}} - \frac{2 \log{\left(\log{\left(x \right)} \right)}}{x^{3} \log{\left(x \right)}} + \frac{2 \log{\left(x + 1 \right)}}{x^{3} \log{\left(x \right)}^{2}} + \frac{2}{x^{3} \log{\left(x \right)}^{2}} + \frac{\log{\left(x + 1 \right)} \log{\left(\log{\left(x \right)} \right)}^{2}}{x^{4}} + \frac{\log{\left(\log{\left(x \right)} \right)}^{2}}{x^{4}} - \frac{2 \log{\left(x + 1 \right)} \log{\left(\log{\left(x \right)} \right)}}{x^{4} \log{\left(x \right)}} - \frac{2 \log{\left(\log{\left(x \right)} \right)}}{x^{4} \log{\left(x \right)}} + \frac{\log{\left(x + 1 \right)}}{x^{4} \log{\left(x \right)}^{2}} + \frac{1}{x^{4} \log{\left(x \right)}^{2}}}{\frac{1}{x^{2} \log{\left(x \right)}} + \frac{1}{x^{2} \log{\left(x \right)}^{2}} - \frac{2 \log{\left(\log{\left(x \right)} \right)}}{x^{3}} + \frac{3}{x^{3} \log{\left(x \right)}} + \frac{1}{x^{3} \log{\left(x \right)}^{2}}}\right)$$
=
$$1$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 2 vez (veces)
Gráfica
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(\frac{\left(x + 1\right) \log{\left(\log{\left(x \right)} \right)}}{x \log{\left(\log{\left(x + 1 \right)} \right)}}\right) = 1$$
$$\lim_{x \to 0^-}\left(\frac{\left(x + 1\right) \log{\left(\log{\left(x \right)} \right)}}{x \log{\left(\log{\left(x + 1 \right)} \right)}}\right) = -\infty$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\left(x + 1\right) \log{\left(\log{\left(x \right)} \right)}}{x \log{\left(\log{\left(x + 1 \right)} \right)}}\right) = -\infty$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{\left(x + 1\right) \log{\left(\log{\left(x \right)} \right)}}{x \log{\left(\log{\left(x + 1 \right)} \right)}}\right) = \infty$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\left(x + 1\right) \log{\left(\log{\left(x \right)} \right)}}{x \log{\left(\log{\left(x + 1 \right)} \right)}}\right) = \infty$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\left(x + 1\right) \log{\left(\log{\left(x \right)} \right)}}{x \log{\left(\log{\left(x + 1 \right)} \right)}}\right) = 1$$
Más detalles con x→-oo
$$\lim_{x \to 0^-}\left(\frac{\left(x + 1\right) \log{\left(\log{\left(x \right)} \right)}}{x \log{\left(\log{\left(x + 1 \right)} \right)}}\right) = -\infty$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\left(x + 1\right) \log{\left(\log{\left(x \right)} \right)}}{x \log{\left(\log{\left(x + 1 \right)} \right)}}\right) = -\infty$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{\left(x + 1\right) \log{\left(\log{\left(x \right)} \right)}}{x \log{\left(\log{\left(x + 1 \right)} \right)}}\right) = \infty$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\left(x + 1\right) \log{\left(\log{\left(x \right)} \right)}}{x \log{\left(\log{\left(x + 1 \right)} \right)}}\right) = \infty$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\left(x + 1\right) \log{\left(\log{\left(x \right)} \right)}}{x \log{\left(\log{\left(x + 1 \right)} \right)}}\right) = 1$$
Más detalles con x→-oo