Sr Examen
Otras calculadoras:
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- Límite de la función:
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Límite de (9+x^2-6*x)/(x^2-3*x)
-
Límite de (-2+sqrt(-2+x))/(-6+x)
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Límite de (sqrt(12+x)-sqrt(4-x))/(-8+x^2+2*x)
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Límite de (sqrt(6+x^2-2*x)-sqrt(-6+x^2+2*x))/(3+x^2-4*x)
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Expresiones idénticas
- (-sqrt(uno +x^ dos)-x*cos(x)+exp(sin(x)))/log(uno -x)^ tres
- ( menos raíz cuadrada de (1 más x al cuadrado ) menos x multiplicar por coseno de (x) más exponente de ( seno de (x))) dividir por logaritmo de (1 menos x) al cubo
- ( menos raíz cuadrada de (uno más x en el grado dos) menos x multiplicar por coseno de (x) más exponente de ( seno de (x))) dividir por logaritmo de (uno menos x) en el grado tres
- (-√(1+x^2)-x*cos(x)+exp(sin(x)))/log(1-x)^3
- (-sqrt(1+x2)-x*cos(x)+exp(sin(x)))/log(1-x)3
- -sqrt1+x2-x*cosx+expsinx/log1-x3
- (-sqrt(1+x²)-x*cos(x)+exp(sin(x)))/log(1-x)³
- (-sqrt(1+x en el grado 2)-x*cos(x)+exp(sin(x)))/log(1-x) en el grado 3
- (-sqrt(1+x^2)-xcos(x)+exp(sin(x)))/log(1-x)^3
- (-sqrt(1+x2)-xcos(x)+exp(sin(x)))/log(1-x)3
- -sqrt1+x2-xcosx+expsinx/log1-x3
- -sqrt1+x^2-xcosx+expsinx/log1-x^3
- (-sqrt(1+x^2)-x*cos(x)+exp(sin(x))) dividir por log(1-x)^3
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Expresiones semejantes
- (sqrt(1+x^2)-x*cos(x)+exp(sin(x)))/log(1-x)^3
- (-sqrt(1+x^2)-x*cos(x)+exp(sin(x)))/log(1+x)^3
- (-sqrt(1+x^2)-x*cos(x)-exp(sin(x)))/log(1-x)^3
- (-sqrt(1-x^2)-x*cos(x)+exp(sin(x)))/log(1-x)^3
- (-sqrt(1+x^2)+x*cos(x)+exp(sin(x)))/log(1-x)^3
- (-sqrt(1+x^2)-x*cosx+exp(sinx))/log(1-x)^3
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Expresiones con funciones
- Raíz cuadrada sqrt
- sqrt((a+x)*(b+x))-x
- sqrt(-1+x+x^2)-sqrt(1+x^2-x)
- sqrt(3+x^2+2*x)-x
- sqrt(x+sqrt(x+sqrt(x)))/sqrt(1+x)
- sqrt(x)-log(x)
- Coseno cos
- cos(3*pi*x)^(1/(x*sin(2*pi*x)))
- cos(1/(pi+x))
- cos(4/x)
- cos(2*x)/tan(x)
- cos(3*x)*sin(2*x)/(cos(2*x)*sin(3*x))
- Exponente exp
- exp(2)/(1+x)
- exp(_x)
- exp(a*sqrt(x)/b)
- exp(3^x+x^3)
- exp(x^2/(-1+x^2))
- Seno sin
- sin(x)^2/(1+cos(x))
- sin(3*x)/(9*x)
- sin(3*x)/x^2
- sin(3*x)/sin(x)
- sin(x)/(x+tan(x))
- Logaritmo log
- log(cos(5*x))/log(cos(4*x))
- log(2*x)*log(-1+2*x)
- log(1+x^2-x)
- log(-3+x^2)/(2+x^2-3*x)
- log(x)/(1+2*log(x)*sin(x))
Límite de la función (-sqrt(1+x^2)-x*cos(x)+exp(sin(x)))/log(1-x)^3
Solución
Ha introducido
[src]
/ ________ \
| / 2 sin(x)|
|- \/ 1 + x - x*cos(x) + e |
lim |----------------------------------|
x->0+| 3 |
\ log (1 - x) /
$$\lim_{x \to 0^+}\left(\frac{\left(- x \cos{\left(x \right)} - \sqrt{x^{2} + 1}\right) + e^{\sin{\left(x \right)}}}{\log{\left(1 - x \right)}^{3}}\right)$$
Limit((-sqrt(1 + x^2) - x*cos(x) + exp(sin(x)))/log(1 - x)^3, x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(- x \cos{\left(x \right)} - \sqrt{x^{2} + 1} + e^{\sin{\left(x \right)}}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \log{\left(1 - x \right)}^{3} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{\left(- x \cos{\left(x \right)} - \sqrt{x^{2} + 1}\right) + e^{\sin{\left(x \right)}}}{\log{\left(1 - x \right)}^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- x \cos{\left(x \right)} - \sqrt{x^{2} + 1} + e^{\sin{\left(x \right)}}\right)}{\frac{d}{d x} \log{\left(1 - x \right)}^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(1 - x\right) \left(x \sin{\left(x \right)} - \frac{x}{\sqrt{x^{2} + 1}} + e^{\sin{\left(x \right)}} \cos{\left(x \right)} - \cos{\left(x \right)}\right)}{3 \log{\left(1 - x \right)}^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{x \sin{\left(x \right)} - \frac{x}{\sqrt{x^{2} + 1}} + e^{\sin{\left(x \right)}} \cos{\left(x \right)} - \cos{\left(x \right)}}{3 \log{\left(1 - x \right)}^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(x \sin{\left(x \right)} - \frac{x}{\sqrt{x^{2} + 1}} + e^{\sin{\left(x \right)}} \cos{\left(x \right)} - \cos{\left(x \right)}\right)}{\frac{d}{d x} \left(- 3 \log{\left(1 - x \right)}^{2}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(1 - x\right) \left(\frac{x^{2}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + x \cos{\left(x \right)} - e^{\sin{\left(x \right)}} \sin{\left(x \right)} + e^{\sin{\left(x \right)}} \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} - \frac{1}{\sqrt{x^{2} + 1}}\right)}{6 \log{\left(1 - x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{x^{2}}{x^{2} \sqrt{x^{2} + 1} + \sqrt{x^{2} + 1}} + x \cos{\left(x \right)} - e^{\sin{\left(x \right)}} \sin{\left(x \right)} + e^{\sin{\left(x \right)}} \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} - \frac{1}{\sqrt{x^{2} + 1}}}{6 \log{\left(1 - x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\frac{x^{2}}{x^{2} \sqrt{x^{2} + 1} + \sqrt{x^{2} + 1}} + x \cos{\left(x \right)} - e^{\sin{\left(x \right)}} \sin{\left(x \right)} + e^{\sin{\left(x \right)}} \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} - \frac{1}{\sqrt{x^{2} + 1}}\right)}{\frac{d}{d x} 6 \log{\left(1 - x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\left(\frac{x}{6} - \frac{1}{6}\right) \left(\frac{x^{2} \left(- \frac{x^{3}}{\sqrt{x^{2} + 1}} - 2 x \sqrt{x^{2} + 1} - \frac{x}{\sqrt{x^{2} + 1}}\right)}{\left(x^{2} \sqrt{x^{2} + 1} + \sqrt{x^{2} + 1}\right)^{2}} - x \sin{\left(x \right)} + \frac{2 x}{x^{2} \sqrt{x^{2} + 1} + \sqrt{x^{2} + 1}} + \frac{x}{\left(x^{2} + 1\right)^{\frac{3}{2}}} - 3 e^{\sin{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)} + e^{\sin{\left(x \right)}} \cos^{3}{\left(x \right)} - e^{\sin{\left(x \right)}} \cos{\left(x \right)} + 3 \cos{\left(x \right)}\right)\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{x^{5}}{6 x^{6} \sqrt{x^{2} + 1} + 18 x^{4} \sqrt{x^{2} + 1} + 18 x^{2} \sqrt{x^{2} + 1} + 6 \sqrt{x^{2} + 1}} + \frac{x^{3} \sqrt{x^{2} + 1}}{3 \left(x^{6} + 3 x^{4} + 3 x^{2} + 1\right)} + \frac{x^{3}}{6 x^{6} \sqrt{x^{2} + 1} + 18 x^{4} \sqrt{x^{2} + 1} + 18 x^{2} \sqrt{x^{2} + 1} + 6 \sqrt{x^{2} + 1}} + \frac{x \sin{\left(x \right)}}{6} - \frac{x}{2 \left(x^{2} \sqrt{x^{2} + 1} + \sqrt{x^{2} + 1}\right)} + \frac{e^{\sin{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)}}{2} - \frac{e^{\sin{\left(x \right)}} \cos^{3}{\left(x \right)}}{6} + \frac{e^{\sin{\left(x \right)}} \cos{\left(x \right)}}{6} - \frac{\cos{\left(x \right)}}{2}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{x^{5}}{6 x^{6} \sqrt{x^{2} + 1} + 18 x^{4} \sqrt{x^{2} + 1} + 18 x^{2} \sqrt{x^{2} + 1} + 6 \sqrt{x^{2} + 1}} + \frac{x^{3} \sqrt{x^{2} + 1}}{3 \left(x^{6} + 3 x^{4} + 3 x^{2} + 1\right)} + \frac{x^{3}}{6 x^{6} \sqrt{x^{2} + 1} + 18 x^{4} \sqrt{x^{2} + 1} + 18 x^{2} \sqrt{x^{2} + 1} + 6 \sqrt{x^{2} + 1}} + \frac{x \sin{\left(x \right)}}{6} - \frac{x}{2 \left(x^{2} \sqrt{x^{2} + 1} + \sqrt{x^{2} + 1}\right)} + \frac{e^{\sin{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)}}{2} - \frac{e^{\sin{\left(x \right)}} \cos^{3}{\left(x \right)}}{6} + \frac{e^{\sin{\left(x \right)}} \cos{\left(x \right)}}{6} - \frac{\cos{\left(x \right)}}{2}\right)$$
=
$$- \frac{1}{2}$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)
0/0,
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(- x \cos{\left(x \right)} - \sqrt{x^{2} + 1} + e^{\sin{\left(x \right)}}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \log{\left(1 - x \right)}^{3} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{\left(- x \cos{\left(x \right)} - \sqrt{x^{2} + 1}\right) + e^{\sin{\left(x \right)}}}{\log{\left(1 - x \right)}^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- x \cos{\left(x \right)} - \sqrt{x^{2} + 1} + e^{\sin{\left(x \right)}}\right)}{\frac{d}{d x} \log{\left(1 - x \right)}^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(1 - x\right) \left(x \sin{\left(x \right)} - \frac{x}{\sqrt{x^{2} + 1}} + e^{\sin{\left(x \right)}} \cos{\left(x \right)} - \cos{\left(x \right)}\right)}{3 \log{\left(1 - x \right)}^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{x \sin{\left(x \right)} - \frac{x}{\sqrt{x^{2} + 1}} + e^{\sin{\left(x \right)}} \cos{\left(x \right)} - \cos{\left(x \right)}}{3 \log{\left(1 - x \right)}^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(x \sin{\left(x \right)} - \frac{x}{\sqrt{x^{2} + 1}} + e^{\sin{\left(x \right)}} \cos{\left(x \right)} - \cos{\left(x \right)}\right)}{\frac{d}{d x} \left(- 3 \log{\left(1 - x \right)}^{2}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(1 - x\right) \left(\frac{x^{2}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + x \cos{\left(x \right)} - e^{\sin{\left(x \right)}} \sin{\left(x \right)} + e^{\sin{\left(x \right)}} \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} - \frac{1}{\sqrt{x^{2} + 1}}\right)}{6 \log{\left(1 - x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{x^{2}}{x^{2} \sqrt{x^{2} + 1} + \sqrt{x^{2} + 1}} + x \cos{\left(x \right)} - e^{\sin{\left(x \right)}} \sin{\left(x \right)} + e^{\sin{\left(x \right)}} \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} - \frac{1}{\sqrt{x^{2} + 1}}}{6 \log{\left(1 - x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\frac{x^{2}}{x^{2} \sqrt{x^{2} + 1} + \sqrt{x^{2} + 1}} + x \cos{\left(x \right)} - e^{\sin{\left(x \right)}} \sin{\left(x \right)} + e^{\sin{\left(x \right)}} \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} - \frac{1}{\sqrt{x^{2} + 1}}\right)}{\frac{d}{d x} 6 \log{\left(1 - x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\left(\frac{x}{6} - \frac{1}{6}\right) \left(\frac{x^{2} \left(- \frac{x^{3}}{\sqrt{x^{2} + 1}} - 2 x \sqrt{x^{2} + 1} - \frac{x}{\sqrt{x^{2} + 1}}\right)}{\left(x^{2} \sqrt{x^{2} + 1} + \sqrt{x^{2} + 1}\right)^{2}} - x \sin{\left(x \right)} + \frac{2 x}{x^{2} \sqrt{x^{2} + 1} + \sqrt{x^{2} + 1}} + \frac{x}{\left(x^{2} + 1\right)^{\frac{3}{2}}} - 3 e^{\sin{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)} + e^{\sin{\left(x \right)}} \cos^{3}{\left(x \right)} - e^{\sin{\left(x \right)}} \cos{\left(x \right)} + 3 \cos{\left(x \right)}\right)\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{x^{5}}{6 x^{6} \sqrt{x^{2} + 1} + 18 x^{4} \sqrt{x^{2} + 1} + 18 x^{2} \sqrt{x^{2} + 1} + 6 \sqrt{x^{2} + 1}} + \frac{x^{3} \sqrt{x^{2} + 1}}{3 \left(x^{6} + 3 x^{4} + 3 x^{2} + 1\right)} + \frac{x^{3}}{6 x^{6} \sqrt{x^{2} + 1} + 18 x^{4} \sqrt{x^{2} + 1} + 18 x^{2} \sqrt{x^{2} + 1} + 6 \sqrt{x^{2} + 1}} + \frac{x \sin{\left(x \right)}}{6} - \frac{x}{2 \left(x^{2} \sqrt{x^{2} + 1} + \sqrt{x^{2} + 1}\right)} + \frac{e^{\sin{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)}}{2} - \frac{e^{\sin{\left(x \right)}} \cos^{3}{\left(x \right)}}{6} + \frac{e^{\sin{\left(x \right)}} \cos{\left(x \right)}}{6} - \frac{\cos{\left(x \right)}}{2}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{x^{5}}{6 x^{6} \sqrt{x^{2} + 1} + 18 x^{4} \sqrt{x^{2} + 1} + 18 x^{2} \sqrt{x^{2} + 1} + 6 \sqrt{x^{2} + 1}} + \frac{x^{3} \sqrt{x^{2} + 1}}{3 \left(x^{6} + 3 x^{4} + 3 x^{2} + 1\right)} + \frac{x^{3}}{6 x^{6} \sqrt{x^{2} + 1} + 18 x^{4} \sqrt{x^{2} + 1} + 18 x^{2} \sqrt{x^{2} + 1} + 6 \sqrt{x^{2} + 1}} + \frac{x \sin{\left(x \right)}}{6} - \frac{x}{2 \left(x^{2} \sqrt{x^{2} + 1} + \sqrt{x^{2} + 1}\right)} + \frac{e^{\sin{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)}}{2} - \frac{e^{\sin{\left(x \right)}} \cos^{3}{\left(x \right)}}{6} + \frac{e^{\sin{\left(x \right)}} \cos{\left(x \right)}}{6} - \frac{\cos{\left(x \right)}}{2}\right)$$
=
$$- \frac{1}{2}$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)
Gráfica
A la izquierda y a la derecha
[src]
/ ________ \
| / 2 sin(x)|
|- \/ 1 + x - x*cos(x) + e |
lim |----------------------------------|
x->0+| 3 |
\ log (1 - x) /
$$\lim_{x \to 0^+}\left(\frac{\left(- x \cos{\left(x \right)} - \sqrt{x^{2} + 1}\right) + e^{\sin{\left(x \right)}}}{\log{\left(1 - x \right)}^{3}}\right)$$
-1/2
$$- \frac{1}{2}$$
= -0.5
/ ________ \
| / 2 sin(x)|
|- \/ 1 + x - x*cos(x) + e |
lim |----------------------------------|
x->0-| 3 |
\ log (1 - x) /
$$\lim_{x \to 0^-}\left(\frac{\left(- x \cos{\left(x \right)} - \sqrt{x^{2} + 1}\right) + e^{\sin{\left(x \right)}}}{\log{\left(1 - x \right)}^{3}}\right)$$
-1/2
$$- \frac{1}{2}$$
= -0.5
= -0.5
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\left(- x \cos{\left(x \right)} - \sqrt{x^{2} + 1}\right) + e^{\sin{\left(x \right)}}}{\log{\left(1 - x \right)}^{3}}\right) = - \frac{1}{2}$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\left(- x \cos{\left(x \right)} - \sqrt{x^{2} + 1}\right) + e^{\sin{\left(x \right)}}}{\log{\left(1 - x \right)}^{3}}\right) = - \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{\left(- x \cos{\left(x \right)} - \sqrt{x^{2} + 1}\right) + e^{\sin{\left(x \right)}}}{\log{\left(1 - x \right)}^{3}}\right) = \left\langle -\infty, 0\right\rangle$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\left(- x \cos{\left(x \right)} - \sqrt{x^{2} + 1}\right) + e^{\sin{\left(x \right)}}}{\log{\left(1 - x \right)}^{3}}\right) = 0$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\left(- x \cos{\left(x \right)} - \sqrt{x^{2} + 1}\right) + e^{\sin{\left(x \right)}}}{\log{\left(1 - x \right)}^{3}}\right) = 0$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\left(- x \cos{\left(x \right)} - \sqrt{x^{2} + 1}\right) + e^{\sin{\left(x \right)}}}{\log{\left(1 - x \right)}^{3}}\right) = \left\langle -\infty, 0\right\rangle$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\left(- x \cos{\left(x \right)} - \sqrt{x^{2} + 1}\right) + e^{\sin{\left(x \right)}}}{\log{\left(1 - x \right)}^{3}}\right) = - \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{\left(- x \cos{\left(x \right)} - \sqrt{x^{2} + 1}\right) + e^{\sin{\left(x \right)}}}{\log{\left(1 - x \right)}^{3}}\right) = \left\langle -\infty, 0\right\rangle$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\left(- x \cos{\left(x \right)} - \sqrt{x^{2} + 1}\right) + e^{\sin{\left(x \right)}}}{\log{\left(1 - x \right)}^{3}}\right) = 0$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\left(- x \cos{\left(x \right)} - \sqrt{x^{2} + 1}\right) + e^{\sin{\left(x \right)}}}{\log{\left(1 - x \right)}^{3}}\right) = 0$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\left(- x \cos{\left(x \right)} - \sqrt{x^{2} + 1}\right) + e^{\sin{\left(x \right)}}}{\log{\left(1 - x \right)}^{3}}\right) = \left\langle -\infty, 0\right\rangle$$
Más detalles con x→-oo