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- Límite de la función:
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Límite de (2-7*x+3*x^2)/(2-5*x+2*x^2)
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Límite de (2-sqrt(x))/(3-sqrt(1+2*x))
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Límite de ((1+tan(x))/(1+sin(x)))^(1/sin(x))
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Límite de (1+x)*(-1+x^3-2*x)/(-5+x^4+4*x^2)
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Expresiones idénticas
- x*(- uno +log(x^ dos)/log((uno +x)^ dos))
- x multiplicar por ( menos 1 más logaritmo de (x al cuadrado ) dividir por logaritmo de ((1 más x) al cuadrado ))
- x multiplicar por ( menos uno más logaritmo de (x en el grado dos) dividir por logaritmo de ((uno más x) en el grado dos))
- x*(-1+log(x2)/log((1+x)2))
- x*-1+logx2/log1+x2
- x*(-1+log(x²)/log((1+x)²))
- x*(-1+log(x en el grado 2)/log((1+x) en el grado 2))
- x(-1+log(x^2)/log((1+x)^2))
- x(-1+log(x2)/log((1+x)2))
- x-1+logx2/log1+x2
- x-1+logx^2/log1+x^2
- x*(-1+log(x^2) dividir por log((1+x)^2))
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Expresiones semejantes
- x*(-1-log(x^2)/log((1+x)^2))
- x*(1+log(x^2)/log((1+x)^2))
- x*(-1+log(x^2)/log((1-x)^2))
-
Expresiones con funciones
- Logaritmo log
- log(cos(3*x))/log(cos(4*x))
- log(-cos(x)+sin(x))
- log(2+sqrt(atan(x)*sin(1/x)))
- log(1+sin(2*x))/sin(3*x)
- log((1+x)/(1-x))/x
- Logaritmo log
- log(cos(3*x))/log(cos(4*x))
- log(-cos(x)+sin(x))
- log(2+sqrt(atan(x)*sin(1/x)))
- log(1+sin(2*x))/sin(3*x)
- log((1+x)/(1-x))/x
Límite de la función x*(-1+log(x^2)/log((1+x)^2))
Solución
Ha introducido
[src]
/ / / 2\ \\
| | log\x / ||
lim |x*|-1 + -------------||
x->oo| | / 2\||
\ \ log\(1 + x) ///
$$\lim_{x \to \infty}\left(x \left(\frac{\log{\left(x^{2} \right)}}{\log{\left(\left(x + 1\right)^{2} \right)}} - 1\right)\right)$$
Limit(x*(-1 + log(x^2)/log((1 + x)^2)), x, oo, dir='-')
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to \infty}\left(\frac{x}{\log{\left(\left(x + 1\right)^{2} \right)}}\right) = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} \frac{1}{\log{\left(x^{2} \right)} - \log{\left(\left(x + 1\right)^{2} \right)}} = -\infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(x \left(\frac{\log{\left(x^{2} \right)}}{\log{\left(\left(x + 1\right)^{2} \right)}} - 1\right)\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \infty}\left(\frac{x \left(\log{\left(x^{2} \right)} - \log{\left(\left(x + 1\right)^{2} \right)}\right)}{\log{\left(\left(x + 1\right)^{2} \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{x}{\log{\left(\left(x + 1\right)^{2} \right)}}}{\frac{d}{d x} \frac{1}{\log{\left(x^{2} \right)} - \log{\left(\left(x + 1\right)^{2} \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- \frac{2 x^{2}}{x^{2} \log{\left(x^{2} + 2 x + 1 \right)}^{2} + 2 x \log{\left(x^{2} + 2 x + 1 \right)}^{2} + \log{\left(x^{2} + 2 x + 1 \right)}^{2}} - \frac{2 x}{x^{2} \log{\left(x^{2} + 2 x + 1 \right)}^{2} + 2 x \log{\left(x^{2} + 2 x + 1 \right)}^{2} + \log{\left(x^{2} + 2 x + 1 \right)}^{2}} + \frac{1}{\log{\left(x^{2} + 2 x + 1 \right)}}}{\frac{2 x}{x^{2} \log{\left(x^{2} \right)}^{2} - 2 x^{2} \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + x^{2} \log{\left(x^{2} + 2 x + 1 \right)}^{2} + 2 x \log{\left(x^{2} \right)}^{2} - 4 x \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + 2 x \log{\left(x^{2} + 2 x + 1 \right)}^{2} + \log{\left(x^{2} \right)}^{2} - 2 \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + \log{\left(x^{2} + 2 x + 1 \right)}^{2}} + \frac{2}{x^{2} \log{\left(x^{2} \right)}^{2} - 2 x^{2} \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + x^{2} \log{\left(x^{2} + 2 x + 1 \right)}^{2} + 2 x \log{\left(x^{2} \right)}^{2} - 4 x \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + 2 x \log{\left(x^{2} + 2 x + 1 \right)}^{2} + \log{\left(x^{2} \right)}^{2} - 2 \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + \log{\left(x^{2} + 2 x + 1 \right)}^{2}} - \frac{2}{x \log{\left(x^{2} \right)}^{2} - 2 x \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + x \log{\left(x^{2} + 2 x + 1 \right)}^{2}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- \frac{2 x^{2}}{x^{2} \log{\left(x^{2} + 2 x + 1 \right)}^{2} + 2 x \log{\left(x^{2} + 2 x + 1 \right)}^{2} + \log{\left(x^{2} + 2 x + 1 \right)}^{2}} - \frac{2 x}{x^{2} \log{\left(x^{2} + 2 x + 1 \right)}^{2} + 2 x \log{\left(x^{2} + 2 x + 1 \right)}^{2} + \log{\left(x^{2} + 2 x + 1 \right)}^{2}} + \frac{1}{\log{\left(x^{2} + 2 x + 1 \right)}}}{\frac{2 x}{x^{2} \log{\left(x^{2} \right)}^{2} - 2 x^{2} \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + x^{2} \log{\left(x^{2} + 2 x + 1 \right)}^{2} + 2 x \log{\left(x^{2} \right)}^{2} - 4 x \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + 2 x \log{\left(x^{2} + 2 x + 1 \right)}^{2} + \log{\left(x^{2} \right)}^{2} - 2 \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + \log{\left(x^{2} + 2 x + 1 \right)}^{2}} + \frac{2}{x^{2} \log{\left(x^{2} \right)}^{2} - 2 x^{2} \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + x^{2} \log{\left(x^{2} + 2 x + 1 \right)}^{2} + 2 x \log{\left(x^{2} \right)}^{2} - 4 x \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + 2 x \log{\left(x^{2} + 2 x + 1 \right)}^{2} + \log{\left(x^{2} \right)}^{2} - 2 \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + \log{\left(x^{2} + 2 x + 1 \right)}^{2}} - \frac{2}{x \log{\left(x^{2} \right)}^{2} - 2 x \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + x \log{\left(x^{2} + 2 x + 1 \right)}^{2}}}\right)$$
=
$$0$$
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
$$\lim_{x \to \infty}\left(\frac{x}{\log{\left(\left(x + 1\right)^{2} \right)}}\right) = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} \frac{1}{\log{\left(x^{2} \right)} - \log{\left(\left(x + 1\right)^{2} \right)}} = -\infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(x \left(\frac{\log{\left(x^{2} \right)}}{\log{\left(\left(x + 1\right)^{2} \right)}} - 1\right)\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \infty}\left(\frac{x \left(\log{\left(x^{2} \right)} - \log{\left(\left(x + 1\right)^{2} \right)}\right)}{\log{\left(\left(x + 1\right)^{2} \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{x}{\log{\left(\left(x + 1\right)^{2} \right)}}}{\frac{d}{d x} \frac{1}{\log{\left(x^{2} \right)} - \log{\left(\left(x + 1\right)^{2} \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- \frac{2 x^{2}}{x^{2} \log{\left(x^{2} + 2 x + 1 \right)}^{2} + 2 x \log{\left(x^{2} + 2 x + 1 \right)}^{2} + \log{\left(x^{2} + 2 x + 1 \right)}^{2}} - \frac{2 x}{x^{2} \log{\left(x^{2} + 2 x + 1 \right)}^{2} + 2 x \log{\left(x^{2} + 2 x + 1 \right)}^{2} + \log{\left(x^{2} + 2 x + 1 \right)}^{2}} + \frac{1}{\log{\left(x^{2} + 2 x + 1 \right)}}}{\frac{2 x}{x^{2} \log{\left(x^{2} \right)}^{2} - 2 x^{2} \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + x^{2} \log{\left(x^{2} + 2 x + 1 \right)}^{2} + 2 x \log{\left(x^{2} \right)}^{2} - 4 x \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + 2 x \log{\left(x^{2} + 2 x + 1 \right)}^{2} + \log{\left(x^{2} \right)}^{2} - 2 \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + \log{\left(x^{2} + 2 x + 1 \right)}^{2}} + \frac{2}{x^{2} \log{\left(x^{2} \right)}^{2} - 2 x^{2} \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + x^{2} \log{\left(x^{2} + 2 x + 1 \right)}^{2} + 2 x \log{\left(x^{2} \right)}^{2} - 4 x \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + 2 x \log{\left(x^{2} + 2 x + 1 \right)}^{2} + \log{\left(x^{2} \right)}^{2} - 2 \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + \log{\left(x^{2} + 2 x + 1 \right)}^{2}} - \frac{2}{x \log{\left(x^{2} \right)}^{2} - 2 x \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + x \log{\left(x^{2} + 2 x + 1 \right)}^{2}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- \frac{2 x^{2}}{x^{2} \log{\left(x^{2} + 2 x + 1 \right)}^{2} + 2 x \log{\left(x^{2} + 2 x + 1 \right)}^{2} + \log{\left(x^{2} + 2 x + 1 \right)}^{2}} - \frac{2 x}{x^{2} \log{\left(x^{2} + 2 x + 1 \right)}^{2} + 2 x \log{\left(x^{2} + 2 x + 1 \right)}^{2} + \log{\left(x^{2} + 2 x + 1 \right)}^{2}} + \frac{1}{\log{\left(x^{2} + 2 x + 1 \right)}}}{\frac{2 x}{x^{2} \log{\left(x^{2} \right)}^{2} - 2 x^{2} \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + x^{2} \log{\left(x^{2} + 2 x + 1 \right)}^{2} + 2 x \log{\left(x^{2} \right)}^{2} - 4 x \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + 2 x \log{\left(x^{2} + 2 x + 1 \right)}^{2} + \log{\left(x^{2} \right)}^{2} - 2 \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + \log{\left(x^{2} + 2 x + 1 \right)}^{2}} + \frac{2}{x^{2} \log{\left(x^{2} \right)}^{2} - 2 x^{2} \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + x^{2} \log{\left(x^{2} + 2 x + 1 \right)}^{2} + 2 x \log{\left(x^{2} \right)}^{2} - 4 x \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + 2 x \log{\left(x^{2} + 2 x + 1 \right)}^{2} + \log{\left(x^{2} \right)}^{2} - 2 \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + \log{\left(x^{2} + 2 x + 1 \right)}^{2}} - \frac{2}{x \log{\left(x^{2} \right)}^{2} - 2 x \log{\left(x^{2} \right)} \log{\left(x^{2} + 2 x + 1 \right)} + x \log{\left(x^{2} + 2 x + 1 \right)}^{2}}}\right)$$
=
$$0$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 1 vez (veces)
Gráfica
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(x \left(\frac{\log{\left(x^{2} \right)}}{\log{\left(\left(x + 1\right)^{2} \right)}} - 1\right)\right) = 0$$
$$\lim_{x \to 0^-}\left(x \left(\frac{\log{\left(x^{2} \right)}}{\log{\left(\left(x + 1\right)^{2} \right)}} - 1\right)\right) = -\infty$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(x \left(\frac{\log{\left(x^{2} \right)}}{\log{\left(\left(x + 1\right)^{2} \right)}} - 1\right)\right) = -\infty$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(x \left(\frac{\log{\left(x^{2} \right)}}{\log{\left(\left(x + 1\right)^{2} \right)}} - 1\right)\right) = -1$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(x \left(\frac{\log{\left(x^{2} \right)}}{\log{\left(\left(x + 1\right)^{2} \right)}} - 1\right)\right) = -1$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(x \left(\frac{\log{\left(x^{2} \right)}}{\log{\left(\left(x + 1\right)^{2} \right)}} - 1\right)\right) = 0$$
Más detalles con x→-oo
$$\lim_{x \to 0^-}\left(x \left(\frac{\log{\left(x^{2} \right)}}{\log{\left(\left(x + 1\right)^{2} \right)}} - 1\right)\right) = -\infty$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(x \left(\frac{\log{\left(x^{2} \right)}}{\log{\left(\left(x + 1\right)^{2} \right)}} - 1\right)\right) = -\infty$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(x \left(\frac{\log{\left(x^{2} \right)}}{\log{\left(\left(x + 1\right)^{2} \right)}} - 1\right)\right) = -1$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(x \left(\frac{\log{\left(x^{2} \right)}}{\log{\left(\left(x + 1\right)^{2} \right)}} - 1\right)\right) = -1$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(x \left(\frac{\log{\left(x^{2} \right)}}{\log{\left(\left(x + 1\right)^{2} \right)}} - 1\right)\right) = 0$$
Más detalles con x→-oo