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Límite de (3+2*n)/(5+3*n)
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Límite de (1+4/x)^(2*x)
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Límite de (x/(-3+x))^(-5+x)
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Límite de (-10+x^2+3*x)/(-2-5*x+3*x^2)
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Expresiones idénticas
- log(uno +x)^(uno +x)- uno /x^ dos
- logaritmo de (1 más x) en el grado (1 más x) menos 1 dividir por x al cuadrado
- logaritmo de (uno más x) en el grado (uno más x) menos uno dividir por x en el grado dos
- log(1+x)(1+x)-1/x2
- log1+x1+x-1/x2
- log(1+x)^(1+x)-1/x²
- log(1+x) en el grado (1+x)-1/x en el grado 2
- log1+x^1+x-1/x^2
- log(1+x)^(1+x)-1 dividir por x^2
-
Expresiones semejantes
- log(1+x)^(1-x)-1/x^2
- log(1-x)^(1+x)-1/x^2
- log(1+x)^(1+x)+1/x^2
-
Expresiones con funciones
- Logaritmo log
- log(1+x^2)/(1-sqrt(1+x^2))
- log(cos(3*x))/log(cos(5*x))
- log(1+sin(x))/sin(x)^4
- log(1+k*x)/x
- log(x)/(1+x^2)
Límite de la función log(1+x)^(1+x)-1/x^2
Solución
Ha introducido
[src]
/ 1 + x 1 \
lim |log (1 + x) - --|
x->oo| 2|
\ x /
$$\lim_{x \to \infty}\left(\log{\left(x + 1 \right)}^{x + 1} - \frac{1}{x^{2}}\right)$$
Limit(log(1 + x)^(1 + x) - 1/x^2, x, oo, dir='-')
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to \infty}\left(x^{2} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x} - 1\right) = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} x^{2} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(\log{\left(x + 1 \right)}^{x + 1} - \frac{1}{x^{2}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \infty}\left(\frac{x^{2} \log{\left(x + 1 \right)}^{x + 1} - 1}{x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(x^{2} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x} - 1\right)}{\frac{d}{d x} x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{x^{3} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}}{x \log{\left(x + 1 \right)} + \log{\left(x + 1 \right)}} + x^{2} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x} \log{\left(\log{\left(x + 1 \right)} \right)} + \frac{x^{2} \log{\left(x + 1 \right)}^{x}}{x + 1} + 2 x \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}}{2 x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(\frac{x^{3} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}}{x \log{\left(x + 1 \right)} + \log{\left(x + 1 \right)}} + x^{2} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x} \log{\left(\log{\left(x + 1 \right)} \right)} + \frac{x^{2} \log{\left(x + 1 \right)}^{x}}{x + 1} + 2 x \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}\right)}{\frac{d}{d x} 2 x}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{x^{4} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}}{2 x^{3} \log{\left(x + 1 \right)}^{2} + 6 x^{2} \log{\left(x + 1 \right)}^{2} + 6 x \log{\left(x + 1 \right)}^{2} + 2 \log{\left(x + 1 \right)}^{2}} + \frac{x^{4} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}}{2 \left(x^{2} \log{\left(x + 1 \right)}^{2} + 2 x \log{\left(x + 1 \right)}^{2} + \log{\left(x + 1 \right)}^{2}\right)} - \frac{x^{3} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}}{2 x^{3} \log{\left(x + 1 \right)}^{2} + 6 x^{2} \log{\left(x + 1 \right)}^{2} + 6 x \log{\left(x + 1 \right)}^{2} + 2 \log{\left(x + 1 \right)}^{2}} - \frac{x^{3} \log{\left(x + 1 \right)}^{2} \log{\left(x + 1 \right)}^{x}}{2 \left(x^{2} \log{\left(x + 1 \right)}^{2} + 2 x \log{\left(x + 1 \right)}^{2} + \log{\left(x + 1 \right)}^{2}\right)} + \frac{2 x^{3} \log{\left(x + 1 \right)}^{x}}{2 x^{2} \log{\left(x + 1 \right)} + 4 x \log{\left(x + 1 \right)} + 2 \log{\left(x + 1 \right)}} + \frac{x^{3} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x} \log{\left(\log{\left(x + 1 \right)} \right)}}{x \log{\left(x + 1 \right)} + \log{\left(x + 1 \right)}} + \frac{x^{2} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x} \log{\left(\log{\left(x + 1 \right)} \right)}^{2}}{2} - \frac{x^{2} \log{\left(x + 1 \right)}^{x}}{2 \left(x^{2} + 2 x + 1\right)} + \frac{5 x^{2} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}}{2 \left(x \log{\left(x + 1 \right)} + \log{\left(x + 1 \right)}\right)} + \frac{x^{2} \log{\left(x + 1 \right)}^{x} \log{\left(\log{\left(x + 1 \right)} \right)}}{x + 1} + \frac{x^{2} \log{\left(x + 1 \right)}^{x}}{2 \left(x + 1\right)} + 2 x \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x} \log{\left(\log{\left(x + 1 \right)} \right)} + \frac{2 x \log{\left(x + 1 \right)}^{x}}{x + 1} + \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{x^{4} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}}{2 x^{3} \log{\left(x + 1 \right)}^{2} + 6 x^{2} \log{\left(x + 1 \right)}^{2} + 6 x \log{\left(x + 1 \right)}^{2} + 2 \log{\left(x + 1 \right)}^{2}} + \frac{x^{4} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}}{2 \left(x^{2} \log{\left(x + 1 \right)}^{2} + 2 x \log{\left(x + 1 \right)}^{2} + \log{\left(x + 1 \right)}^{2}\right)} - \frac{x^{3} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}}{2 x^{3} \log{\left(x + 1 \right)}^{2} + 6 x^{2} \log{\left(x + 1 \right)}^{2} + 6 x \log{\left(x + 1 \right)}^{2} + 2 \log{\left(x + 1 \right)}^{2}} - \frac{x^{3} \log{\left(x + 1 \right)}^{2} \log{\left(x + 1 \right)}^{x}}{2 \left(x^{2} \log{\left(x + 1 \right)}^{2} + 2 x \log{\left(x + 1 \right)}^{2} + \log{\left(x + 1 \right)}^{2}\right)} + \frac{2 x^{3} \log{\left(x + 1 \right)}^{x}}{2 x^{2} \log{\left(x + 1 \right)} + 4 x \log{\left(x + 1 \right)} + 2 \log{\left(x + 1 \right)}} + \frac{x^{3} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x} \log{\left(\log{\left(x + 1 \right)} \right)}}{x \log{\left(x + 1 \right)} + \log{\left(x + 1 \right)}} + \frac{x^{2} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x} \log{\left(\log{\left(x + 1 \right)} \right)}^{2}}{2} - \frac{x^{2} \log{\left(x + 1 \right)}^{x}}{2 \left(x^{2} + 2 x + 1\right)} + \frac{5 x^{2} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}}{2 \left(x \log{\left(x + 1 \right)} + \log{\left(x + 1 \right)}\right)} + \frac{x^{2} \log{\left(x + 1 \right)}^{x} \log{\left(\log{\left(x + 1 \right)} \right)}}{x + 1} + \frac{x^{2} \log{\left(x + 1 \right)}^{x}}{2 \left(x + 1\right)} + 2 x \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x} \log{\left(\log{\left(x + 1 \right)} \right)} + \frac{2 x \log{\left(x + 1 \right)}^{x}}{x + 1} + \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}\right)$$
=
$$\infty$$
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
$$\lim_{x \to \infty}\left(x^{2} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x} - 1\right) = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} x^{2} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(\log{\left(x + 1 \right)}^{x + 1} - \frac{1}{x^{2}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \infty}\left(\frac{x^{2} \log{\left(x + 1 \right)}^{x + 1} - 1}{x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(x^{2} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x} - 1\right)}{\frac{d}{d x} x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{x^{3} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}}{x \log{\left(x + 1 \right)} + \log{\left(x + 1 \right)}} + x^{2} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x} \log{\left(\log{\left(x + 1 \right)} \right)} + \frac{x^{2} \log{\left(x + 1 \right)}^{x}}{x + 1} + 2 x \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}}{2 x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(\frac{x^{3} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}}{x \log{\left(x + 1 \right)} + \log{\left(x + 1 \right)}} + x^{2} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x} \log{\left(\log{\left(x + 1 \right)} \right)} + \frac{x^{2} \log{\left(x + 1 \right)}^{x}}{x + 1} + 2 x \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}\right)}{\frac{d}{d x} 2 x}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{x^{4} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}}{2 x^{3} \log{\left(x + 1 \right)}^{2} + 6 x^{2} \log{\left(x + 1 \right)}^{2} + 6 x \log{\left(x + 1 \right)}^{2} + 2 \log{\left(x + 1 \right)}^{2}} + \frac{x^{4} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}}{2 \left(x^{2} \log{\left(x + 1 \right)}^{2} + 2 x \log{\left(x + 1 \right)}^{2} + \log{\left(x + 1 \right)}^{2}\right)} - \frac{x^{3} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}}{2 x^{3} \log{\left(x + 1 \right)}^{2} + 6 x^{2} \log{\left(x + 1 \right)}^{2} + 6 x \log{\left(x + 1 \right)}^{2} + 2 \log{\left(x + 1 \right)}^{2}} - \frac{x^{3} \log{\left(x + 1 \right)}^{2} \log{\left(x + 1 \right)}^{x}}{2 \left(x^{2} \log{\left(x + 1 \right)}^{2} + 2 x \log{\left(x + 1 \right)}^{2} + \log{\left(x + 1 \right)}^{2}\right)} + \frac{2 x^{3} \log{\left(x + 1 \right)}^{x}}{2 x^{2} \log{\left(x + 1 \right)} + 4 x \log{\left(x + 1 \right)} + 2 \log{\left(x + 1 \right)}} + \frac{x^{3} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x} \log{\left(\log{\left(x + 1 \right)} \right)}}{x \log{\left(x + 1 \right)} + \log{\left(x + 1 \right)}} + \frac{x^{2} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x} \log{\left(\log{\left(x + 1 \right)} \right)}^{2}}{2} - \frac{x^{2} \log{\left(x + 1 \right)}^{x}}{2 \left(x^{2} + 2 x + 1\right)} + \frac{5 x^{2} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}}{2 \left(x \log{\left(x + 1 \right)} + \log{\left(x + 1 \right)}\right)} + \frac{x^{2} \log{\left(x + 1 \right)}^{x} \log{\left(\log{\left(x + 1 \right)} \right)}}{x + 1} + \frac{x^{2} \log{\left(x + 1 \right)}^{x}}{2 \left(x + 1\right)} + 2 x \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x} \log{\left(\log{\left(x + 1 \right)} \right)} + \frac{2 x \log{\left(x + 1 \right)}^{x}}{x + 1} + \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{x^{4} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}}{2 x^{3} \log{\left(x + 1 \right)}^{2} + 6 x^{2} \log{\left(x + 1 \right)}^{2} + 6 x \log{\left(x + 1 \right)}^{2} + 2 \log{\left(x + 1 \right)}^{2}} + \frac{x^{4} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}}{2 \left(x^{2} \log{\left(x + 1 \right)}^{2} + 2 x \log{\left(x + 1 \right)}^{2} + \log{\left(x + 1 \right)}^{2}\right)} - \frac{x^{3} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}}{2 x^{3} \log{\left(x + 1 \right)}^{2} + 6 x^{2} \log{\left(x + 1 \right)}^{2} + 6 x \log{\left(x + 1 \right)}^{2} + 2 \log{\left(x + 1 \right)}^{2}} - \frac{x^{3} \log{\left(x + 1 \right)}^{2} \log{\left(x + 1 \right)}^{x}}{2 \left(x^{2} \log{\left(x + 1 \right)}^{2} + 2 x \log{\left(x + 1 \right)}^{2} + \log{\left(x + 1 \right)}^{2}\right)} + \frac{2 x^{3} \log{\left(x + 1 \right)}^{x}}{2 x^{2} \log{\left(x + 1 \right)} + 4 x \log{\left(x + 1 \right)} + 2 \log{\left(x + 1 \right)}} + \frac{x^{3} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x} \log{\left(\log{\left(x + 1 \right)} \right)}}{x \log{\left(x + 1 \right)} + \log{\left(x + 1 \right)}} + \frac{x^{2} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x} \log{\left(\log{\left(x + 1 \right)} \right)}^{2}}{2} - \frac{x^{2} \log{\left(x + 1 \right)}^{x}}{2 \left(x^{2} + 2 x + 1\right)} + \frac{5 x^{2} \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}}{2 \left(x \log{\left(x + 1 \right)} + \log{\left(x + 1 \right)}\right)} + \frac{x^{2} \log{\left(x + 1 \right)}^{x} \log{\left(\log{\left(x + 1 \right)} \right)}}{x + 1} + \frac{x^{2} \log{\left(x + 1 \right)}^{x}}{2 \left(x + 1\right)} + 2 x \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x} \log{\left(\log{\left(x + 1 \right)} \right)} + \frac{2 x \log{\left(x + 1 \right)}^{x}}{x + 1} + \log{\left(x + 1 \right)} \log{\left(x + 1 \right)}^{x}\right)$$
=
$$\infty$$
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
$$\lim_{x \to \infty}\left(\log{\left(x + 1 \right)}^{x + 1} - \frac{1}{x^{2}}\right) = \infty$$
$$\lim_{x \to 0^-}\left(\log{\left(x + 1 \right)}^{x + 1} - \frac{1}{x^{2}}\right) = -\infty$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\log{\left(x + 1 \right)}^{x + 1} - \frac{1}{x^{2}}\right) = -\infty$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\log{\left(x + 1 \right)}^{x + 1} - \frac{1}{x^{2}}\right) = -1 + \log{\left(2 \right)}^{2}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\log{\left(x + 1 \right)}^{x + 1} - \frac{1}{x^{2}}\right) = -1 + \log{\left(2 \right)}^{2}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\log{\left(x + 1 \right)}^{x + 1} - \frac{1}{x^{2}}\right) = 0$$
Más detalles con x→-oo
$$\lim_{x \to 0^-}\left(\log{\left(x + 1 \right)}^{x + 1} - \frac{1}{x^{2}}\right) = -\infty$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\log{\left(x + 1 \right)}^{x + 1} - \frac{1}{x^{2}}\right) = -\infty$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\log{\left(x + 1 \right)}^{x + 1} - \frac{1}{x^{2}}\right) = -1 + \log{\left(2 \right)}^{2}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\log{\left(x + 1 \right)}^{x + 1} - \frac{1}{x^{2}}\right) = -1 + \log{\left(2 \right)}^{2}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\log{\left(x + 1 \right)}^{x + 1} - \frac{1}{x^{2}}\right) = 0$$
Más detalles con x→-oo