Sr Examen
Otras calculadoras:
-
¿Cómo usar?
- Límite de la función:
-
Límite de ((-3+x)/x)^(x/2)
-
Límite de (-1+4*x+5*x^2)/(-2+x+3*x^2)
-
Límite de (-1+x-2*x^2+2*x^3)/(-3+x^3-x^2+3*x)
-
Límite de (1+4/x)^(1+x)
-
Expresiones idénticas
- log(dos +n)^ dos *(dos +n)/((uno +n)*log(uno +n)^ dos)
- logaritmo de (2 más n) al cuadrado multiplicar por (2 más n) dividir por ((1 más n) multiplicar por logaritmo de (1 más n) al cuadrado )
- logaritmo de (dos más n) en el grado dos multiplicar por (dos más n) dividir por ((uno más n) multiplicar por logaritmo de (uno más n) en el grado dos)
- log(2+n)2*(2+n)/((1+n)*log(1+n)2)
- log2+n2*2+n/1+n*log1+n2
- log(2+n)²*(2+n)/((1+n)*log(1+n)²)
- log(2+n) en el grado 2*(2+n)/((1+n)*log(1+n) en el grado 2)
- log(2+n)^2(2+n)/((1+n)log(1+n)^2)
- log(2+n)2(2+n)/((1+n)log(1+n)2)
- log2+n22+n/1+nlog1+n2
- log2+n^22+n/1+nlog1+n^2
- log(2+n)^2*(2+n) dividir por ((1+n)*log(1+n)^2)
-
Expresiones semejantes
- log(2-n)^2*(2+n)/((1+n)*log(1+n)^2)
- log(2+n)^2*(2+n)/((1+n)*log(1-n)^2)
- log(2+n)^2*(2-n)/((1+n)*log(1+n)^2)
- log(2+n)^2*(2+n)/((1-n)*log(1+n)^2)
-
Expresiones con funciones
- Logaritmo log
- log(-7+4*x)/(-1+3^(-2+x))
- log(-7+2*x)/(-4+x)
- log(1+sin(2*x)^2)/(1-cos(x)^2)
- log(x)^(-2)
- log(1+sin(x))*sin(x)/((1-cos(x))*(-1+e^x))
- Logaritmo log
- log(-7+4*x)/(-1+3^(-2+x))
- log(-7+2*x)/(-4+x)
- log(1+sin(2*x)^2)/(1-cos(x)^2)
- log(x)^(-2)
- log(1+sin(x))*sin(x)/((1-cos(x))*(-1+e^x))
Límite de la función log(2+n)^2*(2+n)/((1+n)*log(1+n)^2)
Solución
Ha introducido
[src]
/ 2 \
|log (2 + n)*(2 + n)|
lim |-------------------|
n->oo| 2 |
\(1 + n)*log (1 + n)/
$$\lim_{n \to \infty}\left(\frac{\left(n + 2\right) \log{\left(n + 2 \right)}^{2}}{\left(n + 1\right) \log{\left(n + 1 \right)}^{2}}\right)$$
Limit((log(2 + n)^2*(2 + n))/(((1 + 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
$$\lim_{n \to \infty} \log{\left(n + 2 \right)}^{2} = \infty$$
y el límite para el denominador es
$$\lim_{n \to \infty}\left(\frac{n \log{\left(n + 1 \right)}^{2}}{n + 2} + \frac{\log{\left(n + 1 \right)}^{2}}{n + 2}\right) = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{n \to \infty}\left(\frac{\left(n + 2\right) \log{\left(n + 2 \right)}^{2}}{\left(n + 1\right) \log{\left(n + 1 \right)}^{2}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{n \to \infty}\left(\frac{\left(n + 2\right) \log{\left(n + 2 \right)}^{2}}{\left(n + 1\right) \log{\left(n + 1 \right)}^{2}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \log{\left(n + 2 \right)}^{2}}{\frac{d}{d n} \left(\frac{n \log{\left(n + 1 \right)}^{2}}{n + 2} + \frac{\log{\left(n + 1 \right)}^{2}}{n + 2}\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{1}{\left(\frac{n}{2 \log{\left(n + 2 \right)}} + \frac{1}{\log{\left(n + 2 \right)}}\right) \left(- \frac{n \log{\left(n + 1 \right)}^{2}}{n^{2} + 4 n + 4} + \frac{2 n \log{\left(n + 1 \right)}}{n^{2} + 3 n + 2} - \frac{\log{\left(n + 1 \right)}^{2}}{n^{2} + 4 n + 4} + \frac{2 \log{\left(n + 1 \right)}}{n^{2} + 3 n + 2} + \frac{\log{\left(n + 1 \right)}^{2}}{n + 2}\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{1}{- \frac{n \log{\left(n + 1 \right)}^{2}}{n^{2} + 4 n + 4} + \frac{2 n \log{\left(n + 1 \right)}}{n^{2} + 3 n + 2} - \frac{\log{\left(n + 1 \right)}^{2}}{n^{2} + 4 n + 4} + \frac{2 \log{\left(n + 1 \right)}}{n^{2} + 3 n + 2} + \frac{\log{\left(n + 1 \right)}^{2}}{n + 2}}}{\frac{d}{d n} \left(\frac{n}{2 \log{\left(n + 2 \right)}} + \frac{1}{\log{\left(n + 2 \right)}}\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{n \left(- 2 n - 4\right) \log{\left(n + 1 \right)}^{2}}{\left(n^{2} + 4 n + 4\right)^{2}} - \frac{2 n \left(- 2 n - 3\right) \log{\left(n + 1 \right)}}{\left(n^{2} + 3 n + 2\right)^{2}} + \frac{2 n \log{\left(n + 1 \right)}}{\left(n + 1\right) \left(n^{2} + 4 n + 4\right)} - \frac{2 n}{\left(n + 1\right) \left(n^{2} + 3 n + 2\right)} + \frac{\left(- 2 n - 4\right) \log{\left(n + 1 \right)}^{2}}{\left(n^{2} + 4 n + 4\right)^{2}} - \frac{2 \left(- 2 n - 3\right) \log{\left(n + 1 \right)}}{\left(n^{2} + 3 n + 2\right)^{2}} + \frac{\log{\left(n + 1 \right)}^{2}}{n^{2} + 4 n + 4} - \frac{2 \log{\left(n + 1 \right)}}{n^{2} + 3 n + 2} + \frac{\log{\left(n + 1 \right)}^{2}}{\left(n + 2\right)^{2}} + \frac{2 \log{\left(n + 1 \right)}}{\left(n + 1\right) \left(n^{2} + 4 n + 4\right)} - \frac{2}{\left(n + 1\right) \left(n^{2} + 3 n + 2\right)} - \frac{2 \log{\left(n + 1 \right)}}{\left(n + 1\right) \left(n + 2\right)}}{\left(- \frac{n}{2 \left(n + 2\right) \log{\left(n + 2 \right)}^{2}} + \frac{1}{2 \log{\left(n + 2 \right)}} - \frac{1}{\left(n + 2\right) \log{\left(n + 2 \right)}^{2}}\right) \left(- \frac{n \log{\left(n + 1 \right)}^{2}}{n^{2} + 4 n + 4} + \frac{2 n \log{\left(n + 1 \right)}}{n^{2} + 3 n + 2} - \frac{\log{\left(n + 1 \right)}^{2}}{n^{2} + 4 n + 4} + \frac{2 \log{\left(n + 1 \right)}}{n^{2} + 3 n + 2} + \frac{\log{\left(n + 1 \right)}^{2}}{n + 2}\right)^{2}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{n \left(- 2 n - 4\right) \log{\left(n + 1 \right)}^{2}}{\left(n^{2} + 4 n + 4\right)^{2}} - \frac{2 n \left(- 2 n - 3\right) \log{\left(n + 1 \right)}}{\left(n^{2} + 3 n + 2\right)^{2}} + \frac{2 n \log{\left(n + 1 \right)}}{\left(n + 1\right) \left(n^{2} + 4 n + 4\right)} - \frac{2 n}{\left(n + 1\right) \left(n^{2} + 3 n + 2\right)} + \frac{\left(- 2 n - 4\right) \log{\left(n + 1 \right)}^{2}}{\left(n^{2} + 4 n + 4\right)^{2}} - \frac{2 \left(- 2 n - 3\right) \log{\left(n + 1 \right)}}{\left(n^{2} + 3 n + 2\right)^{2}} + \frac{\log{\left(n + 1 \right)}^{2}}{n^{2} + 4 n + 4} - \frac{2 \log{\left(n + 1 \right)}}{n^{2} + 3 n + 2} + \frac{\log{\left(n + 1 \right)}^{2}}{\left(n + 2\right)^{2}} + \frac{2 \log{\left(n + 1 \right)}}{\left(n + 1\right) \left(n^{2} + 4 n + 4\right)} - \frac{2}{\left(n + 1\right) \left(n^{2} + 3 n + 2\right)} - \frac{2 \log{\left(n + 1 \right)}}{\left(n + 1\right) \left(n + 2\right)}}{\left(- \frac{n}{2 \left(n + 2\right) \log{\left(n + 2 \right)}^{2}} + \frac{1}{2 \log{\left(n + 2 \right)}} - \frac{1}{\left(n + 2\right) \log{\left(n + 2 \right)}^{2}}\right) \left(- \frac{n \log{\left(n + 1 \right)}^{2}}{n^{2} + 4 n + 4} + \frac{2 n \log{\left(n + 1 \right)}}{n^{2} + 3 n + 2} - \frac{\log{\left(n + 1 \right)}^{2}}{n^{2} + 4 n + 4} + \frac{2 \log{\left(n + 1 \right)}}{n^{2} + 3 n + 2} + \frac{\log{\left(n + 1 \right)}^{2}}{n + 2}\right)^{2}}\right)$$
=
$$1$$
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
$$\lim_{n \to \infty} \log{\left(n + 2 \right)}^{2} = \infty$$
y el límite para el denominador es
$$\lim_{n \to \infty}\left(\frac{n \log{\left(n + 1 \right)}^{2}}{n + 2} + \frac{\log{\left(n + 1 \right)}^{2}}{n + 2}\right) = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{n \to \infty}\left(\frac{\left(n + 2\right) \log{\left(n + 2 \right)}^{2}}{\left(n + 1\right) \log{\left(n + 1 \right)}^{2}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{n \to \infty}\left(\frac{\left(n + 2\right) \log{\left(n + 2 \right)}^{2}}{\left(n + 1\right) \log{\left(n + 1 \right)}^{2}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \log{\left(n + 2 \right)}^{2}}{\frac{d}{d n} \left(\frac{n \log{\left(n + 1 \right)}^{2}}{n + 2} + \frac{\log{\left(n + 1 \right)}^{2}}{n + 2}\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{1}{\left(\frac{n}{2 \log{\left(n + 2 \right)}} + \frac{1}{\log{\left(n + 2 \right)}}\right) \left(- \frac{n \log{\left(n + 1 \right)}^{2}}{n^{2} + 4 n + 4} + \frac{2 n \log{\left(n + 1 \right)}}{n^{2} + 3 n + 2} - \frac{\log{\left(n + 1 \right)}^{2}}{n^{2} + 4 n + 4} + \frac{2 \log{\left(n + 1 \right)}}{n^{2} + 3 n + 2} + \frac{\log{\left(n + 1 \right)}^{2}}{n + 2}\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{1}{- \frac{n \log{\left(n + 1 \right)}^{2}}{n^{2} + 4 n + 4} + \frac{2 n \log{\left(n + 1 \right)}}{n^{2} + 3 n + 2} - \frac{\log{\left(n + 1 \right)}^{2}}{n^{2} + 4 n + 4} + \frac{2 \log{\left(n + 1 \right)}}{n^{2} + 3 n + 2} + \frac{\log{\left(n + 1 \right)}^{2}}{n + 2}}}{\frac{d}{d n} \left(\frac{n}{2 \log{\left(n + 2 \right)}} + \frac{1}{\log{\left(n + 2 \right)}}\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{n \left(- 2 n - 4\right) \log{\left(n + 1 \right)}^{2}}{\left(n^{2} + 4 n + 4\right)^{2}} - \frac{2 n \left(- 2 n - 3\right) \log{\left(n + 1 \right)}}{\left(n^{2} + 3 n + 2\right)^{2}} + \frac{2 n \log{\left(n + 1 \right)}}{\left(n + 1\right) \left(n^{2} + 4 n + 4\right)} - \frac{2 n}{\left(n + 1\right) \left(n^{2} + 3 n + 2\right)} + \frac{\left(- 2 n - 4\right) \log{\left(n + 1 \right)}^{2}}{\left(n^{2} + 4 n + 4\right)^{2}} - \frac{2 \left(- 2 n - 3\right) \log{\left(n + 1 \right)}}{\left(n^{2} + 3 n + 2\right)^{2}} + \frac{\log{\left(n + 1 \right)}^{2}}{n^{2} + 4 n + 4} - \frac{2 \log{\left(n + 1 \right)}}{n^{2} + 3 n + 2} + \frac{\log{\left(n + 1 \right)}^{2}}{\left(n + 2\right)^{2}} + \frac{2 \log{\left(n + 1 \right)}}{\left(n + 1\right) \left(n^{2} + 4 n + 4\right)} - \frac{2}{\left(n + 1\right) \left(n^{2} + 3 n + 2\right)} - \frac{2 \log{\left(n + 1 \right)}}{\left(n + 1\right) \left(n + 2\right)}}{\left(- \frac{n}{2 \left(n + 2\right) \log{\left(n + 2 \right)}^{2}} + \frac{1}{2 \log{\left(n + 2 \right)}} - \frac{1}{\left(n + 2\right) \log{\left(n + 2 \right)}^{2}}\right) \left(- \frac{n \log{\left(n + 1 \right)}^{2}}{n^{2} + 4 n + 4} + \frac{2 n \log{\left(n + 1 \right)}}{n^{2} + 3 n + 2} - \frac{\log{\left(n + 1 \right)}^{2}}{n^{2} + 4 n + 4} + \frac{2 \log{\left(n + 1 \right)}}{n^{2} + 3 n + 2} + \frac{\log{\left(n + 1 \right)}^{2}}{n + 2}\right)^{2}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{n \left(- 2 n - 4\right) \log{\left(n + 1 \right)}^{2}}{\left(n^{2} + 4 n + 4\right)^{2}} - \frac{2 n \left(- 2 n - 3\right) \log{\left(n + 1 \right)}}{\left(n^{2} + 3 n + 2\right)^{2}} + \frac{2 n \log{\left(n + 1 \right)}}{\left(n + 1\right) \left(n^{2} + 4 n + 4\right)} - \frac{2 n}{\left(n + 1\right) \left(n^{2} + 3 n + 2\right)} + \frac{\left(- 2 n - 4\right) \log{\left(n + 1 \right)}^{2}}{\left(n^{2} + 4 n + 4\right)^{2}} - \frac{2 \left(- 2 n - 3\right) \log{\left(n + 1 \right)}}{\left(n^{2} + 3 n + 2\right)^{2}} + \frac{\log{\left(n + 1 \right)}^{2}}{n^{2} + 4 n + 4} - \frac{2 \log{\left(n + 1 \right)}}{n^{2} + 3 n + 2} + \frac{\log{\left(n + 1 \right)}^{2}}{\left(n + 2\right)^{2}} + \frac{2 \log{\left(n + 1 \right)}}{\left(n + 1\right) \left(n^{2} + 4 n + 4\right)} - \frac{2}{\left(n + 1\right) \left(n^{2} + 3 n + 2\right)} - \frac{2 \log{\left(n + 1 \right)}}{\left(n + 1\right) \left(n + 2\right)}}{\left(- \frac{n}{2 \left(n + 2\right) \log{\left(n + 2 \right)}^{2}} + \frac{1}{2 \log{\left(n + 2 \right)}} - \frac{1}{\left(n + 2\right) \log{\left(n + 2 \right)}^{2}}\right) \left(- \frac{n \log{\left(n + 1 \right)}^{2}}{n^{2} + 4 n + 4} + \frac{2 n \log{\left(n + 1 \right)}}{n^{2} + 3 n + 2} - \frac{\log{\left(n + 1 \right)}^{2}}{n^{2} + 4 n + 4} + \frac{2 \log{\left(n + 1 \right)}}{n^{2} + 3 n + 2} + \frac{\log{\left(n + 1 \right)}^{2}}{n + 2}\right)^{2}}\right)$$
=
$$1$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 2 vez (veces)
Gráfica
Otros límites con n→0, -oo, +oo, 1
$$\lim_{n \to \infty}\left(\frac{\left(n + 2\right) \log{\left(n + 2 \right)}^{2}}{\left(n + 1\right) \log{\left(n + 1 \right)}^{2}}\right) = 1$$
$$\lim_{n \to 0^-}\left(\frac{\left(n + 2\right) \log{\left(n + 2 \right)}^{2}}{\left(n + 1\right) \log{\left(n + 1 \right)}^{2}}\right) = \infty$$
Más detalles con n→0 a la izquierda
$$\lim_{n \to 0^+}\left(\frac{\left(n + 2\right) \log{\left(n + 2 \right)}^{2}}{\left(n + 1\right) \log{\left(n + 1 \right)}^{2}}\right) = \infty$$
Más detalles con n→0 a la derecha
$$\lim_{n \to 1^-}\left(\frac{\left(n + 2\right) \log{\left(n + 2 \right)}^{2}}{\left(n + 1\right) \log{\left(n + 1 \right)}^{2}}\right) = \frac{3 \log{\left(3 \right)}^{2}}{2 \log{\left(2 \right)}^{2}}$$
Más detalles con n→1 a la izquierda
$$\lim_{n \to 1^+}\left(\frac{\left(n + 2\right) \log{\left(n + 2 \right)}^{2}}{\left(n + 1\right) \log{\left(n + 1 \right)}^{2}}\right) = \frac{3 \log{\left(3 \right)}^{2}}{2 \log{\left(2 \right)}^{2}}$$
Más detalles con n→1 a la derecha
$$\lim_{n \to -\infty}\left(\frac{\left(n + 2\right) \log{\left(n + 2 \right)}^{2}}{\left(n + 1\right) \log{\left(n + 1 \right)}^{2}}\right) = 1$$
Más detalles con n→-oo
$$\lim_{n \to 0^-}\left(\frac{\left(n + 2\right) \log{\left(n + 2 \right)}^{2}}{\left(n + 1\right) \log{\left(n + 1 \right)}^{2}}\right) = \infty$$
Más detalles con n→0 a la izquierda
$$\lim_{n \to 0^+}\left(\frac{\left(n + 2\right) \log{\left(n + 2 \right)}^{2}}{\left(n + 1\right) \log{\left(n + 1 \right)}^{2}}\right) = \infty$$
Más detalles con n→0 a la derecha
$$\lim_{n \to 1^-}\left(\frac{\left(n + 2\right) \log{\left(n + 2 \right)}^{2}}{\left(n + 1\right) \log{\left(n + 1 \right)}^{2}}\right) = \frac{3 \log{\left(3 \right)}^{2}}{2 \log{\left(2 \right)}^{2}}$$
Más detalles con n→1 a la izquierda
$$\lim_{n \to 1^+}\left(\frac{\left(n + 2\right) \log{\left(n + 2 \right)}^{2}}{\left(n + 1\right) \log{\left(n + 1 \right)}^{2}}\right) = \frac{3 \log{\left(3 \right)}^{2}}{2 \log{\left(2 \right)}^{2}}$$
Más detalles con n→1 a la derecha
$$\lim_{n \to -\infty}\left(\frac{\left(n + 2\right) \log{\left(n + 2 \right)}^{2}}{\left(n + 1\right) \log{\left(n + 1 \right)}^{2}}\right) = 1$$
Más detalles con n→-oo
Gráfico