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(-cos(x)+cos(sin(x)))/x^4
Límite de la función/ cos(x)/ sin(x)/ (-cos(x)+cos(sin(x)))/x^4

Límite de la función (-cos(x)+cos(sin(x)))/x^4

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Solución

Ha introducido [src] 
     /-cos(x) + cos(sin(x))\
 lim |---------------------|
x->0+|           4         |
     \          x          /
lim⁡x→0+(−cos⁡(x)+cos⁡(sin⁡(x))x4)\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
0/0,

tal que el límite para el numerador es
lim⁡x→0+(−cos⁡(x)+cos⁡(sin⁡(x)))=0\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→0+x4=0\lim_{x \to 0^+} x^{4} = 0
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
lim⁡x→0+(−cos⁡(x)+cos⁡(sin⁡(x))x4)\lim_{x \to 0^+}\left(\frac{- \cos{\left(x \right)} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right)
=
lim⁡x→0+(ddx(−cos⁡(x)+cos⁡(sin⁡(x)))ddxx4)\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→0+(sin⁡(x)−sin⁡(sin⁡(x))cos⁡(x)4x3)\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→0+(ddx(sin⁡(x)−sin⁡(sin⁡(x))cos⁡(x))ddx4x3)\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→0+(sin⁡(x)sin⁡(sin⁡(x))−cos⁡2(x)cos⁡(sin⁡(x))+cos⁡(x)12x2)\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→0+(ddx(sin⁡(x)sin⁡(sin⁡(x))−cos⁡2(x)cos⁡(sin⁡(x))+cos⁡(x))ddx12x2)\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→0+(3sin⁡(x)cos⁡(x)cos⁡(sin⁡(x))−sin⁡(x)+sin⁡(sin⁡(x))cos⁡3(x)+sin⁡(sin⁡(x))cos⁡(x)24x)\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→0+(ddx(3sin⁡(x)cos⁡(x)cos⁡(sin⁡(x))−sin⁡(x)+sin⁡(sin⁡(x))cos⁡3(x)+sin⁡(sin⁡(x))cos⁡(x))ddx24x)\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→0+(−sin⁡2(x)cos⁡(sin⁡(x))8−sin⁡(x)sin⁡(sin⁡(x))cos⁡2(x)4−sin⁡(x)sin⁡(sin⁡(x))24+cos⁡4(x)cos⁡(sin⁡(x))24+cos⁡2(x)cos⁡(sin⁡(x))6−cos⁡(x)24)\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→0+(−sin⁡2(x)cos⁡(sin⁡(x))8−sin⁡(x)sin⁡(sin⁡(x))cos⁡2(x)4−sin⁡(x)sin⁡(sin⁡(x))24+cos⁡4(x)cos⁡(sin⁡(x))24+cos⁡2(x)cos⁡(sin⁡(x))6−cos⁡(x)24)\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)
=
16\frac{1}{6}
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 4 vez (veces)
Gráfica
02468-8-6-4-2-10100.00.2
Trazar el gráfico
A la izquierda y a la derecha [src] 
     /-cos(x) + cos(sin(x))\
 lim |---------------------|
x->0+|           4         |
     \          x          /
lim⁡x→0+(−cos⁡(x)+cos⁡(sin⁡(x))x4)\lim_{x \to 0^+}\left(\frac{- \cos{\left(x \right)} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right) 
1/6
16\frac{1}{6} 
= 0.166666666666667
     /-cos(x) + cos(sin(x))\
 lim |---------------------|
x->0-|           4         |
     \          x          /
lim⁡x→0−(−cos⁡(x)+cos⁡(sin⁡(x))x4)\lim_{x \to 0^-}\left(\frac{- \cos{\left(x \right)} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right) 
1/6
16\frac{1}{6} 
= 0.166666666666667
= 0.166666666666667
Otros límites con x→0, -oo, +oo, 1
lim⁡x→0−(−cos⁡(x)+cos⁡(sin⁡(x))x4)=16\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→0+(−cos⁡(x)+cos⁡(sin⁡(x))x4)=16\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→∞(−cos⁡(x)+cos⁡(sin⁡(x))x4)=0\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→1−(−cos⁡(x)+cos⁡(sin⁡(x))x4)=−cos⁡(1)+cos⁡(sin⁡(1))\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→1+(−cos⁡(x)+cos⁡(sin⁡(x))x4)=−cos⁡(1)+cos⁡(sin⁡(1))\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→−∞(−cos⁡(x)+cos⁡(sin⁡(x))x4)=0\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
Respuesta rápida [src] 
1/6
16\frac{1}{6}
Abrir y simplificar
Respuesta numérica [src] 
0.166666666666667
0.166666666666667
Gráfico
Límite de la función (-cos(x)+cos(sin(x)))/x^4
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