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Límite de 4-3*x+2*x^2
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Límite de ((3+x)/(-2+x))^x
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Límite de (-8+x^3)/(-6+x+x^2)
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Expresiones idénticas
- (-cos(x)+cos(sin(x)))/x^ cuatro
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- ( menos coseno de (x) más coseno de ( seno de (x))) dividir por x en el grado cuatro
- (-cos(x)+cos(sin(x)))/x4
- -cosx+cossinx/x4
- (-cos(x)+cos(sin(x)))/x⁴
- -cosx+cossinx/x^4
- (-cos(x)+cos(sin(x))) dividir por x^4
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Expresiones semejantes
- (-cos(x)-cos(sin(x)))/x^4
- (cos(x)+cos(sin(x)))/x^4
- (-cosx+cos(sinx))/x^4
-
Expresiones con funciones
- Coseno cos
- cos(x)*sin(x)
- cos(m/x)^x
- cos(2*x)/sin(3*x)
- cos(x)*log(pi-2*x)
- cos(x)*log(x)/x^2
- Coseno cos
- cos(x)*sin(x)
- cos(m/x)^x
- cos(2*x)/sin(3*x)
- cos(x)*log(pi-2*x)
- cos(x)*log(x)/x^2
- Seno sin
- sin(8*x)/x
- sin(5*x)/(7*x)
- sin(6*x)/tan(2*x)
- sin(-2+x)/(-2+x)
- sin(9*x)*tan(6*x)/(1-cos(10*x))
Límite de la función (-cos(x)+cos(sin(x)))/x^4
Solución
Ha introducido
[src]
/-cos(x) + cos(sin(x))\
lim |---------------------|
x->0+| 4 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{- \cos{\left(x \right)} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right)$$
Limit((-cos(x) + cos(sin(x)))/x^4, x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(- \cos{\left(x \right)} + \cos{\left(\sin{\left(x \right)} \right)}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} x^{4} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{- \cos{\left(x \right)} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \cos{\left(x \right)} + \cos{\left(\sin{\left(x \right)} \right)}\right)}{\frac{d}{d x} x^{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} - \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{4 x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\sin{\left(x \right)} - \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}\right)}{\frac{d}{d x} 4 x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} - \cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \cos{\left(x \right)}}{12 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} - \cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \cos{\left(x \right)}\right)}{\frac{d}{d x} 12 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{3 \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \sin{\left(x \right)} + \sin{\left(\sin{\left(x \right)} \right)} \cos^{3}{\left(x \right)} + \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{24 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(3 \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \sin{\left(x \right)} + \sin{\left(\sin{\left(x \right)} \right)} \cos^{3}{\left(x \right)} + \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}\right)}{\frac{d}{d x} 24 x}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\sin^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{8} - \frac{\sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{4} - \frac{\sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)}}{24} + \frac{\cos^{4}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{24} + \frac{\cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{6} - \frac{\cos{\left(x \right)}}{24}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\sin^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{8} - \frac{\sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{4} - \frac{\sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)}}{24} + \frac{\cos^{4}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{24} + \frac{\cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{6} - \frac{\cos{\left(x \right)}}{24}\right)$$
=
$$\frac{1}{6}$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 4 vez (veces)
0/0,
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(- \cos{\left(x \right)} + \cos{\left(\sin{\left(x \right)} \right)}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} x^{4} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{- \cos{\left(x \right)} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \cos{\left(x \right)} + \cos{\left(\sin{\left(x \right)} \right)}\right)}{\frac{d}{d x} x^{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} - \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{4 x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\sin{\left(x \right)} - \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}\right)}{\frac{d}{d x} 4 x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} - \cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \cos{\left(x \right)}}{12 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} - \cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \cos{\left(x \right)}\right)}{\frac{d}{d x} 12 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{3 \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \sin{\left(x \right)} + \sin{\left(\sin{\left(x \right)} \right)} \cos^{3}{\left(x \right)} + \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{24 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(3 \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \sin{\left(x \right)} + \sin{\left(\sin{\left(x \right)} \right)} \cos^{3}{\left(x \right)} + \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}\right)}{\frac{d}{d x} 24 x}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\sin^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{8} - \frac{\sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{4} - \frac{\sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)}}{24} + \frac{\cos^{4}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{24} + \frac{\cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{6} - \frac{\cos{\left(x \right)}}{24}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\sin^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{8} - \frac{\sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{4} - \frac{\sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)}}{24} + \frac{\cos^{4}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{24} + \frac{\cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{6} - \frac{\cos{\left(x \right)}}{24}\right)$$
=
$$\frac{1}{6}$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 4 vez (veces)
Gráfica
A la izquierda y a la derecha
[src]
/-cos(x) + cos(sin(x))\
lim |---------------------|
x->0+| 4 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{- \cos{\left(x \right)} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right)$$
1/6
$$\frac{1}{6}$$
= 0.166666666666667
/-cos(x) + cos(sin(x))\
lim |---------------------|
x->0-| 4 |
\ x /
$$\lim_{x \to 0^-}\left(\frac{- \cos{\left(x \right)} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right)$$
1/6
$$\frac{1}{6}$$
= 0.166666666666667
= 0.166666666666667
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{- \cos{\left(x \right)} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right) = \frac{1}{6}$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{- \cos{\left(x \right)} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right) = \frac{1}{6}$$
$$\lim_{x \to \infty}\left(\frac{- \cos{\left(x \right)} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right) = 0$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{- \cos{\left(x \right)} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right) = - \cos{\left(1 \right)} + \cos{\left(\sin{\left(1 \right)} \right)}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{- \cos{\left(x \right)} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right) = - \cos{\left(1 \right)} + \cos{\left(\sin{\left(1 \right)} \right)}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{- \cos{\left(x \right)} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right) = 0$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{- \cos{\left(x \right)} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right) = \frac{1}{6}$$
$$\lim_{x \to \infty}\left(\frac{- \cos{\left(x \right)} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right) = 0$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{- \cos{\left(x \right)} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right) = - \cos{\left(1 \right)} + \cos{\left(\sin{\left(1 \right)} \right)}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{- \cos{\left(x \right)} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right) = - \cos{\left(1 \right)} + \cos{\left(\sin{\left(1 \right)} \right)}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{- \cos{\left(x \right)} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right) = 0$$
Más detalles con x→-oo
Gráfico