Sr Examen
Otras calculadoras:
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- Límite de la función:
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Límite de ((3+x)^2+(3-x)^2)/((3-x)^2-(3+x)^2)
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Límite de (-1+(1+x)^5-5*x)/(x^2+x^5)
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Límite de (1+4/x)^x
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Límite de ((-3+x)/x)^(x/2)
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Expresiones idénticas
- factorial(n)^ dos *(uno + cinco ^n)*factorial(dos + dos *n)/((uno + cinco * cinco ^n)*factorial(dos *n)*factorial(uno +n)^ dos)
- factorial(n) al cuadrado multiplicar por (1 más 5 en el grado n) multiplicar por factorial(2 más 2 multiplicar por n) dividir por ((1 más 5 multiplicar por 5 en el grado n) multiplicar por factorial(2 multiplicar por n) multiplicar por factorial(1 más n) al cuadrado )
- factorial(n) en el grado dos multiplicar por (uno más cinco en el grado n) multiplicar por factorial(dos más dos multiplicar por n) dividir por ((uno más cinco multiplicar por cinco en el grado n) multiplicar por factorial(dos multiplicar por n) multiplicar por factorial(uno más n) en el grado dos)
- factorial(n)2*(1+5n)*factorial(2+2*n)/((1+5*5n)*factorial(2*n)*factorial(1+n)2)
- factorialn2*1+5n*factorial2+2*n/1+5*5n*factorial2*n*factorial1+n2
- factorial(n)²*(1+5^n)*factorial(2+2*n)/((1+5*5^n)*factorial(2*n)*factorial(1+n)²)
- factorial(n) en el grado 2*(1+5 en el grado n)*factorial(2+2*n)/((1+5*5 en el grado n)*factorial(2*n)*factorial(1+n) en el grado 2)
- factorial(n)^2(1+5^n)factorial(2+2n)/((1+55^n)factorial(2n)factorial(1+n)^2)
- factorial(n)2(1+5n)factorial(2+2n)/((1+55n)factorial(2n)factorial(1+n)2)
- factorialn21+5nfactorial2+2n/1+55nfactorial2nfactorial1+n2
- factorialn^21+5^nfactorial2+2n/1+55^nfactorial2nfactorial1+n^2
- factorial(n)^2*(1+5^n)*factorial(2+2*n) dividir por ((1+5*5^n)*factorial(2*n)*factorial(1+n)^2)
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Expresiones semejantes
- factorial(n)^2*(1+5^n)*factorial(2+2*n)/((1-5*5^n)*factorial(2*n)*factorial(1+n)^2)
- factorial(n)^2*(1-5^n)*factorial(2+2*n)/((1+5*5^n)*factorial(2*n)*factorial(1+n)^2)
- factorial(n)^2*(1+5^n)*factorial(2+2*n)/((1+5*5^n)*factorial(2*n)*factorial(1-n)^2)
- factorial(n)^2*(1+5^n)*factorial(2-2*n)/((1+5*5^n)*factorial(2*n)*factorial(1+n)^2)
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Expresiones con funciones
- factorial
- factorial(-1+x)
- factorial(a)^(-1/x)*(1+x)
- factorial(-1+n)
- factorial((1+n)^n)/factorial((2+n)^n)
- factorial(1+2*n+factorial(2+2*n))/(3+2*n)
- factorial
- factorial(-1+x)
- factorial(a)^(-1/x)*(1+x)
- factorial(-1+n)
- factorial((1+n)^n)/factorial((2+n)^n)
- factorial(1+2*n+factorial(2+2*n))/(3+2*n)
- factorial
- factorial(-1+x)
- factorial(a)^(-1/x)*(1+x)
- factorial(-1+n)
- factorial((1+n)^n)/factorial((2+n)^n)
- factorial(1+2*n+factorial(2+2*n))/(3+2*n)
- factorial
- factorial(-1+x)
- factorial(a)^(-1/x)*(1+x)
- factorial(-1+n)
- factorial((1+n)^n)/factorial((2+n)^n)
- factorial(1+2*n+factorial(2+2*n))/(3+2*n)
Límite de la función/
factorial(n)^2*(1+5^n)*factorial(2+2*n)/((1+5*5^n)*factorial(2*n)*factorial(1+n)^2)
Límite de la función factorial(n)^2*(1+5^n)*factorial(2+2*n)/((1+5*5^n)*factorial(2*n)*factorial(1+n)^2)
Solución
Ha introducido
[src]
/ 2 / n\ \
| n! *\1 + 5 /*(2 + 2*n)! |
lim |---------------------------|
n->oo|/ n\ 2|
\\1 + 5*5 /*(2*n)!*(1 + n)! /
$$\lim_{n \to \infty}\left(\frac{\left(5^{n} + 1\right) n!^{2} \left(2 n + 2\right)!}{\left(5 \cdot 5^{n} + 1\right) \left(2 n\right)! \left(n + 1\right)!^{2}}\right)$$
Limit(((factorial(n)^2*(1 + 5^n))*factorial(2 + 2*n))/((((1 + 5*5^n)*factorial(2*n))*factorial(1 + n)^2)), n, oo, dir='-')
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{n \to \infty}\left(\frac{\left(5^{n} + 1\right) n!^{2} \left(2 \left(n + 1\right)\right)!}{\left(2 n\right)! \left(n + 1\right)!^{2}}\right) = \infty$$
y el límite para el denominador es
$$\lim_{n \to \infty}\left(5 \cdot 5^{n} + 1\right) = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{n \to \infty}\left(\frac{\left(5^{n} + 1\right) n!^{2} \left(2 n + 2\right)!}{\left(5 \cdot 5^{n} + 1\right) \left(2 n\right)! \left(n + 1\right)!^{2}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{n \to \infty}\left(\frac{\left(5^{n} + 1\right) n!^{2} \left(2 \left(n + 1\right)\right)!}{\left(5 \cdot 5^{n} + 1\right) \left(2 n\right)! \left(n + 1\right)!^{2}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{\left(5^{n} + 1\right) n!^{2} \left(2 \left(n + 1\right)\right)!}{\left(2 n\right)! \left(n + 1\right)!^{2}}}{\frac{d}{d n} \left(5 \cdot 5^{n} + 1\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{5^{- n} \left(\frac{5^{n} \log{\left(5 \right)} n!^{2} \left(2 n + 2\right)!}{\left(2 n\right)! \left(n + 1\right)!^{2}} + \frac{2 \cdot 5^{n} n!^{2} \Gamma\left(2 n + 3\right) \operatorname{polygamma}{\left(0,2 n + 3 \right)}}{\left(2 n\right)! \left(n + 1\right)!^{2}} - \frac{2 \cdot 5^{n} n!^{2} \left(2 n + 2\right)! \Gamma\left(n + 2\right) \operatorname{polygamma}{\left(0,n + 2 \right)}}{\left(2 n\right)! \left(n + 1\right)!^{3}} - \frac{2 \cdot 5^{n} n!^{2} \left(2 n + 2\right)! \Gamma\left(2 n + 1\right) \operatorname{polygamma}{\left(0,2 n + 1 \right)}}{\left(2 n\right)!^{2} \left(n + 1\right)!^{2}} + \frac{2 \cdot 5^{n} n! \left(2 n + 2\right)! \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{\left(2 n\right)! \left(n + 1\right)!^{2}} + \frac{2 n!^{2} \Gamma\left(2 n + 3\right) \operatorname{polygamma}{\left(0,2 n + 3 \right)}}{\left(2 n\right)! \left(n + 1\right)!^{2}} - \frac{2 n!^{2} \left(2 n + 2\right)! \Gamma\left(n + 2\right) \operatorname{polygamma}{\left(0,n + 2 \right)}}{\left(2 n\right)! \left(n + 1\right)!^{3}} - \frac{2 n!^{2} \left(2 n + 2\right)! \Gamma\left(2 n + 1\right) \operatorname{polygamma}{\left(0,2 n + 1 \right)}}{\left(2 n\right)!^{2} \left(n + 1\right)!^{2}} + \frac{2 n! \left(2 n + 2\right)! \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{\left(2 n\right)! \left(n + 1\right)!^{2}}\right)}{5 \log{\left(5 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{5^{- n} \left(\frac{5^{n} \log{\left(5 \right)} n!^{2} \left(2 n + 2\right)!}{\left(2 n\right)! \left(n + 1\right)!^{2}} + \frac{2 \cdot 5^{n} n!^{2} \Gamma\left(2 n + 3\right) \operatorname{polygamma}{\left(0,2 n + 3 \right)}}{\left(2 n\right)! \left(n + 1\right)!^{2}} - \frac{2 \cdot 5^{n} n!^{2} \left(2 n + 2\right)! \Gamma\left(n + 2\right) \operatorname{polygamma}{\left(0,n + 2 \right)}}{\left(2 n\right)! \left(n + 1\right)!^{3}} - \frac{2 \cdot 5^{n} n!^{2} \left(2 n + 2\right)! \Gamma\left(2 n + 1\right) \operatorname{polygamma}{\left(0,2 n + 1 \right)}}{\left(2 n\right)!^{2} \left(n + 1\right)!^{2}} + \frac{2 \cdot 5^{n} n! \left(2 n + 2\right)! \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{\left(2 n\right)! \left(n + 1\right)!^{2}} + \frac{2 n!^{2} \Gamma\left(2 n + 3\right) \operatorname{polygamma}{\left(0,2 n + 3 \right)}}{\left(2 n\right)! \left(n + 1\right)!^{2}} - \frac{2 n!^{2} \left(2 n + 2\right)! \Gamma\left(n + 2\right) \operatorname{polygamma}{\left(0,n + 2 \right)}}{\left(2 n\right)! \left(n + 1\right)!^{3}} - \frac{2 n!^{2} \left(2 n + 2\right)! \Gamma\left(2 n + 1\right) \operatorname{polygamma}{\left(0,2 n + 1 \right)}}{\left(2 n\right)!^{2} \left(n + 1\right)!^{2}} + \frac{2 n! \left(2 n + 2\right)! \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{\left(2 n\right)! \left(n + 1\right)!^{2}}\right)}{5 \log{\left(5 \right)}}\right)$$
=
$$\frac{4}{5}$$
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
$$\lim_{n \to \infty}\left(\frac{\left(5^{n} + 1\right) n!^{2} \left(2 \left(n + 1\right)\right)!}{\left(2 n\right)! \left(n + 1\right)!^{2}}\right) = \infty$$
y el límite para el denominador es
$$\lim_{n \to \infty}\left(5 \cdot 5^{n} + 1\right) = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{n \to \infty}\left(\frac{\left(5^{n} + 1\right) n!^{2} \left(2 n + 2\right)!}{\left(5 \cdot 5^{n} + 1\right) \left(2 n\right)! \left(n + 1\right)!^{2}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{n \to \infty}\left(\frac{\left(5^{n} + 1\right) n!^{2} \left(2 \left(n + 1\right)\right)!}{\left(5 \cdot 5^{n} + 1\right) \left(2 n\right)! \left(n + 1\right)!^{2}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{\left(5^{n} + 1\right) n!^{2} \left(2 \left(n + 1\right)\right)!}{\left(2 n\right)! \left(n + 1\right)!^{2}}}{\frac{d}{d n} \left(5 \cdot 5^{n} + 1\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{5^{- n} \left(\frac{5^{n} \log{\left(5 \right)} n!^{2} \left(2 n + 2\right)!}{\left(2 n\right)! \left(n + 1\right)!^{2}} + \frac{2 \cdot 5^{n} n!^{2} \Gamma\left(2 n + 3\right) \operatorname{polygamma}{\left(0,2 n + 3 \right)}}{\left(2 n\right)! \left(n + 1\right)!^{2}} - \frac{2 \cdot 5^{n} n!^{2} \left(2 n + 2\right)! \Gamma\left(n + 2\right) \operatorname{polygamma}{\left(0,n + 2 \right)}}{\left(2 n\right)! \left(n + 1\right)!^{3}} - \frac{2 \cdot 5^{n} n!^{2} \left(2 n + 2\right)! \Gamma\left(2 n + 1\right) \operatorname{polygamma}{\left(0,2 n + 1 \right)}}{\left(2 n\right)!^{2} \left(n + 1\right)!^{2}} + \frac{2 \cdot 5^{n} n! \left(2 n + 2\right)! \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{\left(2 n\right)! \left(n + 1\right)!^{2}} + \frac{2 n!^{2} \Gamma\left(2 n + 3\right) \operatorname{polygamma}{\left(0,2 n + 3 \right)}}{\left(2 n\right)! \left(n + 1\right)!^{2}} - \frac{2 n!^{2} \left(2 n + 2\right)! \Gamma\left(n + 2\right) \operatorname{polygamma}{\left(0,n + 2 \right)}}{\left(2 n\right)! \left(n + 1\right)!^{3}} - \frac{2 n!^{2} \left(2 n + 2\right)! \Gamma\left(2 n + 1\right) \operatorname{polygamma}{\left(0,2 n + 1 \right)}}{\left(2 n\right)!^{2} \left(n + 1\right)!^{2}} + \frac{2 n! \left(2 n + 2\right)! \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{\left(2 n\right)! \left(n + 1\right)!^{2}}\right)}{5 \log{\left(5 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{5^{- n} \left(\frac{5^{n} \log{\left(5 \right)} n!^{2} \left(2 n + 2\right)!}{\left(2 n\right)! \left(n + 1\right)!^{2}} + \frac{2 \cdot 5^{n} n!^{2} \Gamma\left(2 n + 3\right) \operatorname{polygamma}{\left(0,2 n + 3 \right)}}{\left(2 n\right)! \left(n + 1\right)!^{2}} - \frac{2 \cdot 5^{n} n!^{2} \left(2 n + 2\right)! \Gamma\left(n + 2\right) \operatorname{polygamma}{\left(0,n + 2 \right)}}{\left(2 n\right)! \left(n + 1\right)!^{3}} - \frac{2 \cdot 5^{n} n!^{2} \left(2 n + 2\right)! \Gamma\left(2 n + 1\right) \operatorname{polygamma}{\left(0,2 n + 1 \right)}}{\left(2 n\right)!^{2} \left(n + 1\right)!^{2}} + \frac{2 \cdot 5^{n} n! \left(2 n + 2\right)! \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{\left(2 n\right)! \left(n + 1\right)!^{2}} + \frac{2 n!^{2} \Gamma\left(2 n + 3\right) \operatorname{polygamma}{\left(0,2 n + 3 \right)}}{\left(2 n\right)! \left(n + 1\right)!^{2}} - \frac{2 n!^{2} \left(2 n + 2\right)! \Gamma\left(n + 2\right) \operatorname{polygamma}{\left(0,n + 2 \right)}}{\left(2 n\right)! \left(n + 1\right)!^{3}} - \frac{2 n!^{2} \left(2 n + 2\right)! \Gamma\left(2 n + 1\right) \operatorname{polygamma}{\left(0,2 n + 1 \right)}}{\left(2 n\right)!^{2} \left(n + 1\right)!^{2}} + \frac{2 n! \left(2 n + 2\right)! \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{\left(2 n\right)! \left(n + 1\right)!^{2}}\right)}{5 \log{\left(5 \right)}}\right)$$
=
$$\frac{4}{5}$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 1 vez (veces)
Gráfica
Otros límites con n→0, -oo, +oo, 1
$$\lim_{n \to \infty}\left(\frac{\left(5^{n} + 1\right) n!^{2} \left(2 n + 2\right)!}{\left(5 \cdot 5^{n} + 1\right) \left(2 n\right)! \left(n + 1\right)!^{2}}\right) = \frac{4}{5}$$
$$\lim_{n \to 0^-}\left(\frac{\left(5^{n} + 1\right) n!^{2} \left(2 n + 2\right)!}{\left(5 \cdot 5^{n} + 1\right) \left(2 n\right)! \left(n + 1\right)!^{2}}\right) = \frac{2}{3}$$
Más detalles con n→0 a la izquierda
$$\lim_{n \to 0^+}\left(\frac{\left(5^{n} + 1\right) n!^{2} \left(2 n + 2\right)!}{\left(5 \cdot 5^{n} + 1\right) \left(2 n\right)! \left(n + 1\right)!^{2}}\right) = \frac{2}{3}$$
Más detalles con n→0 a la derecha
$$\lim_{n \to 1^-}\left(\frac{\left(5^{n} + 1\right) n!^{2} \left(2 n + 2\right)!}{\left(5 \cdot 5^{n} + 1\right) \left(2 n\right)! \left(n + 1\right)!^{2}}\right) = \frac{9}{13}$$
Más detalles con n→1 a la izquierda
$$\lim_{n \to 1^+}\left(\frac{\left(5^{n} + 1\right) n!^{2} \left(2 n + 2\right)!}{\left(5 \cdot 5^{n} + 1\right) \left(2 n\right)! \left(n + 1\right)!^{2}}\right) = \frac{9}{13}$$
Más detalles con n→1 a la derecha
$$\lim_{n \to -\infty}\left(\frac{\left(5^{n} + 1\right) n!^{2} \left(2 n + 2\right)!}{\left(5 \cdot 5^{n} + 1\right) \left(2 n\right)! \left(n + 1\right)!^{2}}\right) = 1$$
Más detalles con n→-oo
$$\lim_{n \to 0^-}\left(\frac{\left(5^{n} + 1\right) n!^{2} \left(2 n + 2\right)!}{\left(5 \cdot 5^{n} + 1\right) \left(2 n\right)! \left(n + 1\right)!^{2}}\right) = \frac{2}{3}$$
Más detalles con n→0 a la izquierda
$$\lim_{n \to 0^+}\left(\frac{\left(5^{n} + 1\right) n!^{2} \left(2 n + 2\right)!}{\left(5 \cdot 5^{n} + 1\right) \left(2 n\right)! \left(n + 1\right)!^{2}}\right) = \frac{2}{3}$$
Más detalles con n→0 a la derecha
$$\lim_{n \to 1^-}\left(\frac{\left(5^{n} + 1\right) n!^{2} \left(2 n + 2\right)!}{\left(5 \cdot 5^{n} + 1\right) \left(2 n\right)! \left(n + 1\right)!^{2}}\right) = \frac{9}{13}$$
Más detalles con n→1 a la izquierda
$$\lim_{n \to 1^+}\left(\frac{\left(5^{n} + 1\right) n!^{2} \left(2 n + 2\right)!}{\left(5 \cdot 5^{n} + 1\right) \left(2 n\right)! \left(n + 1\right)!^{2}}\right) = \frac{9}{13}$$
Más detalles con n→1 a la derecha
$$\lim_{n \to -\infty}\left(\frac{\left(5^{n} + 1\right) n!^{2} \left(2 n + 2\right)!}{\left(5 \cdot 5^{n} + 1\right) \left(2 n\right)! \left(n + 1\right)!^{2}}\right) = 1$$
Más detalles con n→-oo