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Otras calculadoras:
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- Límite de la función:
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Límite de (-9+x^2)/(3+x)
-
Límite de x^2/(1-cos(6*x))
-
Límite de (-3+sqrt(1+2*x))/(sqrt(-2+x)-sqrt(2))
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Límite de (4+x^2-5*x)/(8+x^2-6*x)
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Expresiones idénticas
- log(n)^ dos *(uno +n)/(n*log(uno +n)^ dos)
- logaritmo de (n) al cuadrado multiplicar por (1 más n) dividir por (n multiplicar por logaritmo de (1 más n) al cuadrado )
- logaritmo de (n) en el grado dos multiplicar por (uno más n) dividir por (n multiplicar por logaritmo de (uno más n) en el grado dos)
- log(n)2*(1+n)/(n*log(1+n)2)
- logn2*1+n/n*log1+n2
- log(n)²*(1+n)/(n*log(1+n)²)
- log(n) en el grado 2*(1+n)/(n*log(1+n) en el grado 2)
- log(n)^2(1+n)/(nlog(1+n)^2)
- log(n)2(1+n)/(nlog(1+n)2)
- logn21+n/nlog1+n2
- logn^21+n/nlog1+n^2
- log(n)^2*(1+n) dividir por (n*log(1+n)^2)
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Expresiones semejantes
- log(n)^2*(1-n)/(n*log(1+n)^2)
- log(n)^2*(1+n)/(n*log(1-n)^2)
-
Expresiones con funciones
- Logaritmo log
- log(sin(x))*tan(x)
- log(e+x)^(1/x)
- log(1+2*x)/x
- log((1+x)/(-1+x))
- log(cos(2*x))/log(cos(3*x))
- Logaritmo log
- log(sin(x))*tan(x)
- log(e+x)^(1/x)
- log(1+2*x)/x
- log((1+x)/(-1+x))
- log(cos(2*x))/log(cos(3*x))
Límite de la función log(n)^2*(1+n)/(n*log(1+n)^2)
Solución
Ha introducido
[src]
/ 2 \
|log (n)*(1 + n)|
lim |---------------|
n->oo| 2 |
\ n*log (1 + n) /
$$\lim_{n \to \infty}\left(\frac{\left(n + 1\right) \log{\left(n \right)}^{2}}{n \log{\left(n + 1 \right)}^{2}}\right)$$
Limit((log(n)^2*(1 + n))/((n*log(1 + n)^2)), n, oo, dir='-')
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{n \to \infty}\left(\frac{\left(n + 1\right) \log{\left(n \right)}^{2}}{n}\right) = \infty$$
y el límite para el denominador es
$$\lim_{n \to \infty} \log{\left(n + 1 \right)}^{2} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{n \to \infty}\left(\frac{\left(n + 1\right) \log{\left(n \right)}^{2}}{n \log{\left(n + 1 \right)}^{2}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{n \to \infty}\left(\frac{\left(n + 1\right) \log{\left(n \right)}^{2}}{n \log{\left(n + 1 \right)}^{2}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{\left(n + 1\right) \log{\left(n \right)}^{2}}{n}}{\frac{d}{d n} \log{\left(n + 1 \right)}^{2}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\left(n + 1\right) \left(\frac{\log{\left(n \right)}^{2}}{n} - \frac{\left(n + 1\right) \log{\left(n \right)}^{2}}{n^{2}} + \frac{2 \left(n + 1\right) \log{\left(n \right)}}{n^{2}}\right)}{2 \log{\left(n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{n + 1}{2 \log{\left(n + 1 \right)}}}{\frac{d}{d n} \frac{1}{\frac{\log{\left(n \right)}^{2}}{n} - \frac{\left(n + 1\right) \log{\left(n \right)}^{2}}{n^{2}} + \frac{2 \left(n + 1\right) \log{\left(n \right)}}{n^{2}}}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{2 \log{\left(n \right)}^{2}}{n^{2} \log{\left(n + 1 \right)}} - \frac{2 \log{\left(n \right)}^{2}}{n^{2} \log{\left(n + 1 \right)}^{2}} - \frac{2 \log{\left(n \right)}^{3}}{n^{3} \log{\left(n + 1 \right)}} + \frac{2 \log{\left(n \right)}^{3}}{n^{3} \log{\left(n + 1 \right)}^{2}} + \frac{4 \log{\left(n \right)}^{2}}{n^{3} \log{\left(n + 1 \right)}} - \frac{4 \log{\left(n \right)}^{2}}{n^{3} \log{\left(n + 1 \right)}^{2}} + \frac{\log{\left(n \right)}^{4}}{2 n^{4} \log{\left(n + 1 \right)}} - \frac{\log{\left(n \right)}^{4}}{2 n^{4} \log{\left(n + 1 \right)}^{2}} - \frac{2 \log{\left(n \right)}^{3}}{n^{4} \log{\left(n + 1 \right)}} + \frac{2 \log{\left(n \right)}^{3}}{n^{4} \log{\left(n + 1 \right)}^{2}} + \frac{2 \log{\left(n \right)}^{2}}{n^{4} \log{\left(n + 1 \right)}} - \frac{2 \log{\left(n \right)}^{2}}{n^{4} \log{\left(n + 1 \right)}^{2}}}{\frac{2 \log{\left(n \right)}}{n^{2}} - \frac{2}{n^{2}} - \frac{2 \log{\left(n \right)}^{2}}{n^{3}} + \frac{6 \log{\left(n \right)}}{n^{3}} - \frac{2}{n^{3}}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{2 \log{\left(n \right)}^{2}}{n^{2} \log{\left(n + 1 \right)}} - \frac{2 \log{\left(n \right)}^{2}}{n^{2} \log{\left(n + 1 \right)}^{2}} - \frac{2 \log{\left(n \right)}^{3}}{n^{3} \log{\left(n + 1 \right)}} + \frac{2 \log{\left(n \right)}^{3}}{n^{3} \log{\left(n + 1 \right)}^{2}} + \frac{4 \log{\left(n \right)}^{2}}{n^{3} \log{\left(n + 1 \right)}} - \frac{4 \log{\left(n \right)}^{2}}{n^{3} \log{\left(n + 1 \right)}^{2}} + \frac{\log{\left(n \right)}^{4}}{2 n^{4} \log{\left(n + 1 \right)}} - \frac{\log{\left(n \right)}^{4}}{2 n^{4} \log{\left(n + 1 \right)}^{2}} - \frac{2 \log{\left(n \right)}^{3}}{n^{4} \log{\left(n + 1 \right)}} + \frac{2 \log{\left(n \right)}^{3}}{n^{4} \log{\left(n + 1 \right)}^{2}} + \frac{2 \log{\left(n \right)}^{2}}{n^{4} \log{\left(n + 1 \right)}} - \frac{2 \log{\left(n \right)}^{2}}{n^{4} \log{\left(n + 1 \right)}^{2}}}{\frac{2 \log{\left(n \right)}}{n^{2}} - \frac{2}{n^{2}} - \frac{2 \log{\left(n \right)}^{2}}{n^{3}} + \frac{6 \log{\left(n \right)}}{n^{3}} - \frac{2}{n^{3}}}\right)$$
=
$$1$$
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
$$\lim_{n \to \infty}\left(\frac{\left(n + 1\right) \log{\left(n \right)}^{2}}{n}\right) = \infty$$
y el límite para el denominador es
$$\lim_{n \to \infty} \log{\left(n + 1 \right)}^{2} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{n \to \infty}\left(\frac{\left(n + 1\right) \log{\left(n \right)}^{2}}{n \log{\left(n + 1 \right)}^{2}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{n \to \infty}\left(\frac{\left(n + 1\right) \log{\left(n \right)}^{2}}{n \log{\left(n + 1 \right)}^{2}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{\left(n + 1\right) \log{\left(n \right)}^{2}}{n}}{\frac{d}{d n} \log{\left(n + 1 \right)}^{2}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\left(n + 1\right) \left(\frac{\log{\left(n \right)}^{2}}{n} - \frac{\left(n + 1\right) \log{\left(n \right)}^{2}}{n^{2}} + \frac{2 \left(n + 1\right) \log{\left(n \right)}}{n^{2}}\right)}{2 \log{\left(n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{n + 1}{2 \log{\left(n + 1 \right)}}}{\frac{d}{d n} \frac{1}{\frac{\log{\left(n \right)}^{2}}{n} - \frac{\left(n + 1\right) \log{\left(n \right)}^{2}}{n^{2}} + \frac{2 \left(n + 1\right) \log{\left(n \right)}}{n^{2}}}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{2 \log{\left(n \right)}^{2}}{n^{2} \log{\left(n + 1 \right)}} - \frac{2 \log{\left(n \right)}^{2}}{n^{2} \log{\left(n + 1 \right)}^{2}} - \frac{2 \log{\left(n \right)}^{3}}{n^{3} \log{\left(n + 1 \right)}} + \frac{2 \log{\left(n \right)}^{3}}{n^{3} \log{\left(n + 1 \right)}^{2}} + \frac{4 \log{\left(n \right)}^{2}}{n^{3} \log{\left(n + 1 \right)}} - \frac{4 \log{\left(n \right)}^{2}}{n^{3} \log{\left(n + 1 \right)}^{2}} + \frac{\log{\left(n \right)}^{4}}{2 n^{4} \log{\left(n + 1 \right)}} - \frac{\log{\left(n \right)}^{4}}{2 n^{4} \log{\left(n + 1 \right)}^{2}} - \frac{2 \log{\left(n \right)}^{3}}{n^{4} \log{\left(n + 1 \right)}} + \frac{2 \log{\left(n \right)}^{3}}{n^{4} \log{\left(n + 1 \right)}^{2}} + \frac{2 \log{\left(n \right)}^{2}}{n^{4} \log{\left(n + 1 \right)}} - \frac{2 \log{\left(n \right)}^{2}}{n^{4} \log{\left(n + 1 \right)}^{2}}}{\frac{2 \log{\left(n \right)}}{n^{2}} - \frac{2}{n^{2}} - \frac{2 \log{\left(n \right)}^{2}}{n^{3}} + \frac{6 \log{\left(n \right)}}{n^{3}} - \frac{2}{n^{3}}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{2 \log{\left(n \right)}^{2}}{n^{2} \log{\left(n + 1 \right)}} - \frac{2 \log{\left(n \right)}^{2}}{n^{2} \log{\left(n + 1 \right)}^{2}} - \frac{2 \log{\left(n \right)}^{3}}{n^{3} \log{\left(n + 1 \right)}} + \frac{2 \log{\left(n \right)}^{3}}{n^{3} \log{\left(n + 1 \right)}^{2}} + \frac{4 \log{\left(n \right)}^{2}}{n^{3} \log{\left(n + 1 \right)}} - \frac{4 \log{\left(n \right)}^{2}}{n^{3} \log{\left(n + 1 \right)}^{2}} + \frac{\log{\left(n \right)}^{4}}{2 n^{4} \log{\left(n + 1 \right)}} - \frac{\log{\left(n \right)}^{4}}{2 n^{4} \log{\left(n + 1 \right)}^{2}} - \frac{2 \log{\left(n \right)}^{3}}{n^{4} \log{\left(n + 1 \right)}} + \frac{2 \log{\left(n \right)}^{3}}{n^{4} \log{\left(n + 1 \right)}^{2}} + \frac{2 \log{\left(n \right)}^{2}}{n^{4} \log{\left(n + 1 \right)}} - \frac{2 \log{\left(n \right)}^{2}}{n^{4} \log{\left(n + 1 \right)}^{2}}}{\frac{2 \log{\left(n \right)}}{n^{2}} - \frac{2}{n^{2}} - \frac{2 \log{\left(n \right)}^{2}}{n^{3}} + \frac{6 \log{\left(n \right)}}{n^{3}} - \frac{2}{n^{3}}}\right)$$
=
$$1$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 2 vez (veces)
Gráfica
Otros límites con n→0, -oo, +oo, 1
$$\lim_{n \to \infty}\left(\frac{\left(n + 1\right) \log{\left(n \right)}^{2}}{n \log{\left(n + 1 \right)}^{2}}\right) = 1$$
$$\lim_{n \to 0^-}\left(\frac{\left(n + 1\right) \log{\left(n \right)}^{2}}{n \log{\left(n + 1 \right)}^{2}}\right) = -\infty$$
Más detalles con n→0 a la izquierda
$$\lim_{n \to 0^+}\left(\frac{\left(n + 1\right) \log{\left(n \right)}^{2}}{n \log{\left(n + 1 \right)}^{2}}\right) = \infty$$
Más detalles con n→0 a la derecha
$$\lim_{n \to 1^-}\left(\frac{\left(n + 1\right) \log{\left(n \right)}^{2}}{n \log{\left(n + 1 \right)}^{2}}\right) = 0$$
Más detalles con n→1 a la izquierda
$$\lim_{n \to 1^+}\left(\frac{\left(n + 1\right) \log{\left(n \right)}^{2}}{n \log{\left(n + 1 \right)}^{2}}\right) = 0$$
Más detalles con n→1 a la derecha
$$\lim_{n \to -\infty}\left(\frac{\left(n + 1\right) \log{\left(n \right)}^{2}}{n \log{\left(n + 1 \right)}^{2}}\right) = 1$$
Más detalles con n→-oo
$$\lim_{n \to 0^-}\left(\frac{\left(n + 1\right) \log{\left(n \right)}^{2}}{n \log{\left(n + 1 \right)}^{2}}\right) = -\infty$$
Más detalles con n→0 a la izquierda
$$\lim_{n \to 0^+}\left(\frac{\left(n + 1\right) \log{\left(n \right)}^{2}}{n \log{\left(n + 1 \right)}^{2}}\right) = \infty$$
Más detalles con n→0 a la derecha
$$\lim_{n \to 1^-}\left(\frac{\left(n + 1\right) \log{\left(n \right)}^{2}}{n \log{\left(n + 1 \right)}^{2}}\right) = 0$$
Más detalles con n→1 a la izquierda
$$\lim_{n \to 1^+}\left(\frac{\left(n + 1\right) \log{\left(n \right)}^{2}}{n \log{\left(n + 1 \right)}^{2}}\right) = 0$$
Más detalles con n→1 a la derecha
$$\lim_{n \to -\infty}\left(\frac{\left(n + 1\right) \log{\left(n \right)}^{2}}{n \log{\left(n + 1 \right)}^{2}}\right) = 1$$
Más detalles con n→-oo
Gráfico