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- Límite de la función:
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Límite de x/(-2+x)
-
Límite de x^(-2)
-
Límite de (-4+x^2)/(6+x^2-5*x)
-
Límite de (2-sqrt(-3+x))/(-49+x^2)
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Expresiones idénticas
- x^(- dos)-x-x^ dos +log(|x|)
- x en el grado ( menos 2) menos x menos x al cuadrado más logaritmo de ( módulo de x|)
- x en el grado ( menos dos) menos x menos x en el grado dos más logaritmo de ( módulo de x|)
- x(-2)-x-x2+log(|x|)
- x-2-x-x2+log|x|
- x^(-2)-x-x²+log(|x|)
- x en el grado (-2)-x-x en el grado 2+log(|x|)
- x^-2-x-x^2+log|x|
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Expresiones semejantes
- x^(-2)-x+x^2+log(|x|)
- x^(-2)-x-x^2-log(|x|)
- x^(2)-x-x^2+log(|x|)
- x^(-2)+x-x^2+log(|x|)
-
Expresiones con funciones
- Logaritmo log
- log(1+x)/x
- log(x)^(1/x)
- log(-1+x)
- log(1-7*x)/sin(pi*(7+x))
- log(1+x^2)
- Módulo |
- |-4+x|/(-4+x)
- |x|/x
- |x|
- |x^3+6*x^2+9*x|/x
- |-3+x|/(-3+x)
Límite de la función x^(-2)-x-x^2+log(|x|)
Solución
Ha introducido
[src]
/1 2 \
lim |-- - x - x + log(|x|)|
x->oo| 2 |
\x /
$$\lim_{x \to \infty}\left(\left(- x^{2} + \left(- x + \frac{1}{x^{2}}\right)\right) + \log{\left(\left|{x}\right| \right)}\right)$$
Limit(x^(-2) - x - x^2 + log(|x|), x, oo, dir='-')
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to \infty}\left(- x^{4} - x^{3} + x^{2} \log{\left(\left|{x}\right| \right)} + 1\right) = -\infty$$
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.
$$\lim_{x \to \infty}\left(\left(- x^{2} + \left(- x + \frac{1}{x^{2}}\right)\right) + \log{\left(\left|{x}\right| \right)}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \infty}\left(\frac{- x^{4} - x^{3} + x^{2} \log{\left(\left|{x}\right| \right)} + 1}{x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- x^{4} - x^{3} + x^{2} \log{\left(\left|{x}\right| \right)} + 1\right)}{\frac{d}{d x} x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- 4 x^{3} - 3 x^{2} + 2 x \log{\left(\left|{x}\right| \right)} + \frac{x \operatorname{re}{\left(x\right)} \operatorname{sign}{\left(x \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\left|{x}\right|} + \frac{x \operatorname{im}{\left(x\right)} \operatorname{sign}{\left(x \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\left|{x}\right|}}{2 x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- 4 x^{3} - 3 x^{2} + 2 x \log{\left(\left|{x}\right| \right)} + \frac{x \operatorname{re}{\left(x\right)} \operatorname{sign}{\left(x \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\left|{x}\right|} + \frac{x \operatorname{im}{\left(x\right)} \operatorname{sign}{\left(x \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\left|{x}\right|}\right)}{\frac{d}{d x} 2 x}\right)$$
=
$$\lim_{x \to \infty}\left(- 6 x^{2} + \frac{x \operatorname{re}{\left(x\right)} \operatorname{sign}{\left(x \right)} \frac{d^{2}}{d x^{2}} \operatorname{re}{\left(x\right)}}{2 \left|{x}\right|} + \frac{x \operatorname{re}{\left(x\right)} \frac{d}{d x} \operatorname{re}{\left(x\right)} \frac{d}{d x} \operatorname{sign}{\left(x \right)}}{2 \left|{x}\right|} + \frac{x \operatorname{im}{\left(x\right)} \operatorname{sign}{\left(x \right)} \frac{d^{2}}{d x^{2}} \operatorname{im}{\left(x\right)}}{2 \left|{x}\right|} + \frac{x \operatorname{im}{\left(x\right)} \frac{d}{d x} \operatorname{im}{\left(x\right)} \frac{d}{d x} \operatorname{sign}{\left(x \right)}}{2 \left|{x}\right|} - 3 x + \frac{x \operatorname{sign}{\left(x \right)} \left(\frac{d}{d x} \operatorname{re}{\left(x\right)}\right)^{2}}{2 \left|{x}\right|} + \frac{x \operatorname{sign}{\left(x \right)} \left(\frac{d}{d x} \operatorname{im}{\left(x\right)}\right)^{2}}{2 \left|{x}\right|} + \log{\left(\left|{x}\right| \right)} - \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} \operatorname{sign}^{2}{\left(x \right)} \left(\frac{d}{d x} \operatorname{re}{\left(x\right)}\right)^{2}}{2 \left|{x}\right|^{2}} - \frac{\operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} \operatorname{sign}^{2}{\left(x \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\left|{x}\right|^{2}} + \frac{3 \operatorname{re}{\left(x\right)} \operatorname{sign}{\left(x \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{2 \left|{x}\right|} - \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2} \operatorname{sign}^{2}{\left(x \right)} \left(\frac{d}{d x} \operatorname{im}{\left(x\right)}\right)^{2}}{2 \left|{x}\right|^{2}} + \frac{3 \operatorname{im}{\left(x\right)} \operatorname{sign}{\left(x \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{2 \left|{x}\right|}\right)$$
=
$$\lim_{x \to \infty}\left(- 6 x^{2} + \frac{x \operatorname{re}{\left(x\right)} \operatorname{sign}{\left(x \right)} \frac{d^{2}}{d x^{2}} \operatorname{re}{\left(x\right)}}{2 \left|{x}\right|} + \frac{x \operatorname{re}{\left(x\right)} \frac{d}{d x} \operatorname{re}{\left(x\right)} \frac{d}{d x} \operatorname{sign}{\left(x \right)}}{2 \left|{x}\right|} + \frac{x \operatorname{im}{\left(x\right)} \operatorname{sign}{\left(x \right)} \frac{d^{2}}{d x^{2}} \operatorname{im}{\left(x\right)}}{2 \left|{x}\right|} + \frac{x \operatorname{im}{\left(x\right)} \frac{d}{d x} \operatorname{im}{\left(x\right)} \frac{d}{d x} \operatorname{sign}{\left(x \right)}}{2 \left|{x}\right|} - 3 x + \frac{x \operatorname{sign}{\left(x \right)} \left(\frac{d}{d x} \operatorname{re}{\left(x\right)}\right)^{2}}{2 \left|{x}\right|} + \frac{x \operatorname{sign}{\left(x \right)} \left(\frac{d}{d x} \operatorname{im}{\left(x\right)}\right)^{2}}{2 \left|{x}\right|} + \log{\left(\left|{x}\right| \right)} - \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} \operatorname{sign}^{2}{\left(x \right)} \left(\frac{d}{d x} \operatorname{re}{\left(x\right)}\right)^{2}}{2 \left|{x}\right|^{2}} - \frac{\operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} \operatorname{sign}^{2}{\left(x \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\left|{x}\right|^{2}} + \frac{3 \operatorname{re}{\left(x\right)} \operatorname{sign}{\left(x \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{2 \left|{x}\right|} - \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2} \operatorname{sign}^{2}{\left(x \right)} \left(\frac{d}{d x} \operatorname{im}{\left(x\right)}\right)^{2}}{2 \left|{x}\right|^{2}} + \frac{3 \operatorname{im}{\left(x\right)} \operatorname{sign}{\left(x \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{2 \left|{x}\right|}\right)$$
=
$$-\infty$$
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
$$\lim_{x \to \infty}\left(- x^{4} - x^{3} + x^{2} \log{\left(\left|{x}\right| \right)} + 1\right) = -\infty$$
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.
$$\lim_{x \to \infty}\left(\left(- x^{2} + \left(- x + \frac{1}{x^{2}}\right)\right) + \log{\left(\left|{x}\right| \right)}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \infty}\left(\frac{- x^{4} - x^{3} + x^{2} \log{\left(\left|{x}\right| \right)} + 1}{x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- x^{4} - x^{3} + x^{2} \log{\left(\left|{x}\right| \right)} + 1\right)}{\frac{d}{d x} x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- 4 x^{3} - 3 x^{2} + 2 x \log{\left(\left|{x}\right| \right)} + \frac{x \operatorname{re}{\left(x\right)} \operatorname{sign}{\left(x \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\left|{x}\right|} + \frac{x \operatorname{im}{\left(x\right)} \operatorname{sign}{\left(x \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\left|{x}\right|}}{2 x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- 4 x^{3} - 3 x^{2} + 2 x \log{\left(\left|{x}\right| \right)} + \frac{x \operatorname{re}{\left(x\right)} \operatorname{sign}{\left(x \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\left|{x}\right|} + \frac{x \operatorname{im}{\left(x\right)} \operatorname{sign}{\left(x \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\left|{x}\right|}\right)}{\frac{d}{d x} 2 x}\right)$$
=
$$\lim_{x \to \infty}\left(- 6 x^{2} + \frac{x \operatorname{re}{\left(x\right)} \operatorname{sign}{\left(x \right)} \frac{d^{2}}{d x^{2}} \operatorname{re}{\left(x\right)}}{2 \left|{x}\right|} + \frac{x \operatorname{re}{\left(x\right)} \frac{d}{d x} \operatorname{re}{\left(x\right)} \frac{d}{d x} \operatorname{sign}{\left(x \right)}}{2 \left|{x}\right|} + \frac{x \operatorname{im}{\left(x\right)} \operatorname{sign}{\left(x \right)} \frac{d^{2}}{d x^{2}} \operatorname{im}{\left(x\right)}}{2 \left|{x}\right|} + \frac{x \operatorname{im}{\left(x\right)} \frac{d}{d x} \operatorname{im}{\left(x\right)} \frac{d}{d x} \operatorname{sign}{\left(x \right)}}{2 \left|{x}\right|} - 3 x + \frac{x \operatorname{sign}{\left(x \right)} \left(\frac{d}{d x} \operatorname{re}{\left(x\right)}\right)^{2}}{2 \left|{x}\right|} + \frac{x \operatorname{sign}{\left(x \right)} \left(\frac{d}{d x} \operatorname{im}{\left(x\right)}\right)^{2}}{2 \left|{x}\right|} + \log{\left(\left|{x}\right| \right)} - \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} \operatorname{sign}^{2}{\left(x \right)} \left(\frac{d}{d x} \operatorname{re}{\left(x\right)}\right)^{2}}{2 \left|{x}\right|^{2}} - \frac{\operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} \operatorname{sign}^{2}{\left(x \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\left|{x}\right|^{2}} + \frac{3 \operatorname{re}{\left(x\right)} \operatorname{sign}{\left(x \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{2 \left|{x}\right|} - \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2} \operatorname{sign}^{2}{\left(x \right)} \left(\frac{d}{d x} \operatorname{im}{\left(x\right)}\right)^{2}}{2 \left|{x}\right|^{2}} + \frac{3 \operatorname{im}{\left(x\right)} \operatorname{sign}{\left(x \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{2 \left|{x}\right|}\right)$$
=
$$\lim_{x \to \infty}\left(- 6 x^{2} + \frac{x \operatorname{re}{\left(x\right)} \operatorname{sign}{\left(x \right)} \frac{d^{2}}{d x^{2}} \operatorname{re}{\left(x\right)}}{2 \left|{x}\right|} + \frac{x \operatorname{re}{\left(x\right)} \frac{d}{d x} \operatorname{re}{\left(x\right)} \frac{d}{d x} \operatorname{sign}{\left(x \right)}}{2 \left|{x}\right|} + \frac{x \operatorname{im}{\left(x\right)} \operatorname{sign}{\left(x \right)} \frac{d^{2}}{d x^{2}} \operatorname{im}{\left(x\right)}}{2 \left|{x}\right|} + \frac{x \operatorname{im}{\left(x\right)} \frac{d}{d x} \operatorname{im}{\left(x\right)} \frac{d}{d x} \operatorname{sign}{\left(x \right)}}{2 \left|{x}\right|} - 3 x + \frac{x \operatorname{sign}{\left(x \right)} \left(\frac{d}{d x} \operatorname{re}{\left(x\right)}\right)^{2}}{2 \left|{x}\right|} + \frac{x \operatorname{sign}{\left(x \right)} \left(\frac{d}{d x} \operatorname{im}{\left(x\right)}\right)^{2}}{2 \left|{x}\right|} + \log{\left(\left|{x}\right| \right)} - \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} \operatorname{sign}^{2}{\left(x \right)} \left(\frac{d}{d x} \operatorname{re}{\left(x\right)}\right)^{2}}{2 \left|{x}\right|^{2}} - \frac{\operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} \operatorname{sign}^{2}{\left(x \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\left|{x}\right|^{2}} + \frac{3 \operatorname{re}{\left(x\right)} \operatorname{sign}{\left(x \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{2 \left|{x}\right|} - \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2} \operatorname{sign}^{2}{\left(x \right)} \left(\frac{d}{d x} \operatorname{im}{\left(x\right)}\right)^{2}}{2 \left|{x}\right|^{2}} + \frac{3 \operatorname{im}{\left(x\right)} \operatorname{sign}{\left(x \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{2 \left|{x}\right|}\right)$$
=
$$-\infty$$
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
$$\lim_{x \to \infty}\left(\left(- x^{2} + \left(- x + \frac{1}{x^{2}}\right)\right) + \log{\left(\left|{x}\right| \right)}\right) = -\infty$$
$$\lim_{x \to 0^-}\left(\left(- x^{2} + \left(- x + \frac{1}{x^{2}}\right)\right) + \log{\left(\left|{x}\right| \right)}\right) = \infty$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\left(- x^{2} + \left(- x + \frac{1}{x^{2}}\right)\right) + \log{\left(\left|{x}\right| \right)}\right) = \infty$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\left(- x^{2} + \left(- x + \frac{1}{x^{2}}\right)\right) + \log{\left(\left|{x}\right| \right)}\right) = -1$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\left(- x^{2} + \left(- x + \frac{1}{x^{2}}\right)\right) + \log{\left(\left|{x}\right| \right)}\right) = -1$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\left(- x^{2} + \left(- x + \frac{1}{x^{2}}\right)\right) + \log{\left(\left|{x}\right| \right)}\right) = -\infty$$
Más detalles con x→-oo
$$\lim_{x \to 0^-}\left(\left(- x^{2} + \left(- x + \frac{1}{x^{2}}\right)\right) + \log{\left(\left|{x}\right| \right)}\right) = \infty$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\left(- x^{2} + \left(- x + \frac{1}{x^{2}}\right)\right) + \log{\left(\left|{x}\right| \right)}\right) = \infty$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\left(- x^{2} + \left(- x + \frac{1}{x^{2}}\right)\right) + \log{\left(\left|{x}\right| \right)}\right) = -1$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\left(- x^{2} + \left(- x + \frac{1}{x^{2}}\right)\right) + \log{\left(\left|{x}\right| \right)}\right) = -1$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\left(- x^{2} + \left(- x + \frac{1}{x^{2}}\right)\right) + \log{\left(\left|{x}\right| \right)}\right) = -\infty$$
Más detalles con x→-oo