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-x/(x-sin(x))+tan(x)
Límite de la función/ sin(x)/ tan(x)/ -x/(x-sin(x))+tan(x)

Límite de la función -x/(x-sin(x))+tan(x)

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Solución

Ha introducido [src] 
     /   -x              \
 lim |---------- + tan(x)|
x->0+\x - sin(x)         /
lim⁡x→0+((−1)xx−sin⁡(x)+tan⁡(x))\lim_{x \to 0^+}\left(\frac{\left(-1\right) x}{x - \sin{\left(x \right)}} + \tan{\left(x \right)}\right) 
Limit((-x)/(x - sin(x)) + tan(x), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
0/0,

tal que el límite para el numerador es
lim⁡x→0+(xtan⁡(x)−x−sin⁡(x)tan⁡(x))=0\lim_{x \to 0^+}\left(x \tan{\left(x \right)} - x - \sin{\left(x \right)} \tan{\left(x \right)}\right) = 0
y el límite para el denominador es
lim⁡x→0+(x−sin⁡(x))=0\lim_{x \to 0^+}\left(x - \sin{\left(x \right)}\right) = 0
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
lim⁡x→0+((−1)xx−sin⁡(x)+tan⁡(x))\lim_{x \to 0^+}\left(\frac{\left(-1\right) x}{x - \sin{\left(x \right)}} + \tan{\left(x \right)}\right)
=
Introducimos una pequeña modificación de la función bajo el signo del límite
lim⁡x→0+(−x+(x−sin⁡(x))tan⁡(x)x−sin⁡(x))\lim_{x \to 0^+}\left(\frac{- x + \left(x - \sin{\left(x \right)}\right) \tan{\left(x \right)}}{x - \sin{\left(x \right)}}\right)
=
lim⁡x→0+(ddx(xtan⁡(x)−x−sin⁡(x)tan⁡(x))ddx(x−sin⁡(x)))\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(x \tan{\left(x \right)} - x - \sin{\left(x \right)} \tan{\left(x \right)}\right)}{\frac{d}{d x} \left(x - \sin{\left(x \right)}\right)}\right)
=
lim⁡x→0+(xtan⁡2(x)+x−sin⁡(x)tan⁡2(x)−sin⁡(x)−cos⁡(x)tan⁡(x)+tan⁡(x)−11−cos⁡(x))\lim_{x \to 0^+}\left(\frac{x \tan^{2}{\left(x \right)} + x - \sin{\left(x \right)} \tan^{2}{\left(x \right)} - \sin{\left(x \right)} - \cos{\left(x \right)} \tan{\left(x \right)} + \tan{\left(x \right)} - 1}{1 - \cos{\left(x \right)}}\right)
=
lim⁡x→0+(ddx(xtan⁡2(x)+x−sin⁡(x)tan⁡2(x)−sin⁡(x)−cos⁡(x)tan⁡(x)+tan⁡(x)−1)ddx(1−cos⁡(x)))\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(x \tan^{2}{\left(x \right)} + x - \sin{\left(x \right)} \tan^{2}{\left(x \right)} - \sin{\left(x \right)} - \cos{\left(x \right)} \tan{\left(x \right)} + \tan{\left(x \right)} - 1\right)}{\frac{d}{d x} \left(1 - \cos{\left(x \right)}\right)}\right)
=
lim⁡x→0+(2xtan⁡3(x)+2xtan⁡(x)−2sin⁡(x)tan⁡3(x)−sin⁡(x)tan⁡(x)−2cos⁡(x)tan⁡2(x)−2cos⁡(x)+2tan⁡2(x)+2sin⁡(x))\lim_{x \to 0^+}\left(\frac{2 x \tan^{3}{\left(x \right)} + 2 x \tan{\left(x \right)} - 2 \sin{\left(x \right)} \tan^{3}{\left(x \right)} - \sin{\left(x \right)} \tan{\left(x \right)} - 2 \cos{\left(x \right)} \tan^{2}{\left(x \right)} - 2 \cos{\left(x \right)} + 2 \tan^{2}{\left(x \right)} + 2}{\sin{\left(x \right)}}\right)
=
lim⁡x→0+(ddx(2xtan⁡3(x)+2xtan⁡(x)−2sin⁡(x)tan⁡3(x)−sin⁡(x)tan⁡(x)−2cos⁡(x)tan⁡2(x)−2cos⁡(x)+2tan⁡2(x)+2)ddxsin⁡(x))\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(2 x \tan^{3}{\left(x \right)} + 2 x \tan{\left(x \right)} - 2 \sin{\left(x \right)} \tan^{3}{\left(x \right)} - \sin{\left(x \right)} \tan{\left(x \right)} - 2 \cos{\left(x \right)} \tan^{2}{\left(x \right)} - 2 \cos{\left(x \right)} + 2 \tan^{2}{\left(x \right)} + 2\right)}{\frac{d}{d x} \sin{\left(x \right)}}\right)
=
lim⁡x→0+(6xtan⁡4(x)+8xtan⁡2(x)+2x−6sin⁡(x)tan⁡4(x)−5sin⁡(x)tan⁡2(x)+sin⁡(x)−6cos⁡(x)tan⁡3(x)−5cos⁡(x)tan⁡(x)+6tan⁡3(x)+6tan⁡(x)cos⁡(x))\lim_{x \to 0^+}\left(\frac{6 x \tan^{4}{\left(x \right)} + 8 x \tan^{2}{\left(x \right)} + 2 x - 6 \sin{\left(x \right)} \tan^{4}{\left(x \right)} - 5 \sin{\left(x \right)} \tan^{2}{\left(x \right)} + \sin{\left(x \right)} - 6 \cos{\left(x \right)} \tan^{3}{\left(x \right)} - 5 \cos{\left(x \right)} \tan{\left(x \right)} + 6 \tan^{3}{\left(x \right)} + 6 \tan{\left(x \right)}}{\cos{\left(x \right)}}\right)
=
lim⁡x→0+(6xtan⁡4(x)+8xtan⁡2(x)+2x−6sin⁡(x)tan⁡4(x)−5sin⁡(x)tan⁡2(x)+sin⁡(x)−6cos⁡(x)tan⁡3(x)−5cos⁡(x)tan⁡(x)+6tan⁡3(x)+6tan⁡(x)cos⁡(x))\lim_{x \to 0^+}\left(\frac{6 x \tan^{4}{\left(x \right)} + 8 x \tan^{2}{\left(x \right)} + 2 x - 6 \sin{\left(x \right)} \tan^{4}{\left(x \right)} - 5 \sin{\left(x \right)} \tan^{2}{\left(x \right)} + \sin{\left(x \right)} - 6 \cos{\left(x \right)} \tan^{3}{\left(x \right)} - 5 \cos{\left(x \right)} \tan{\left(x \right)} + 6 \tan^{3}{\left(x \right)} + 6 \tan{\left(x \right)}}{\cos{\left(x \right)}}\right)
=
−∞-\infty
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)
Gráfica
02468-8-6-4-2-1010-200000200000
Trazar el gráfico
Respuesta rápida [src] 
-oo
−∞-\infty
Abrir y simplificar
A la izquierda y a la derecha [src] 
     /   -x              \
 lim |---------- + tan(x)|
x->0+\x - sin(x)         /
lim⁡x→0+((−1)xx−sin⁡(x)+tan⁡(x))\lim_{x \to 0^+}\left(\frac{\left(-1\right) x}{x - \sin{\left(x \right)}} + \tan{\left(x \right)}\right) 
-oo
−∞-\infty 
= -136806.293377731
     /   -x              \
 lim |---------- + tan(x)|
x->0-\x - sin(x)         /
lim⁡x→0−((−1)xx−sin⁡(x)+tan⁡(x))\lim_{x \to 0^-}\left(\frac{\left(-1\right) x}{x - \sin{\left(x \right)}} + \tan{\left(x \right)}\right) 
-oo
−∞-\infty 
= -136806.306622958
= -136806.306622958
Otros límites con x→0, -oo, +oo, 1
lim⁡x→0−((−1)xx−sin⁡(x)+tan⁡(x))=−∞\lim_{x \to 0^-}\left(\frac{\left(-1\right) x}{x - \sin{\left(x \right)}} + \tan{\left(x \right)}\right) = -\infty
Más detalles con x→0 a la izquierda
lim⁡x→0+((−1)xx−sin⁡(x)+tan⁡(x))=−∞\lim_{x \to 0^+}\left(\frac{\left(-1\right) x}{x - \sin{\left(x \right)}} + \tan{\left(x \right)}\right) = -\infty
lim⁡x→∞((−1)xx−sin⁡(x)+tan⁡(x))\lim_{x \to \infty}\left(\frac{\left(-1\right) x}{x - \sin{\left(x \right)}} + \tan{\left(x \right)}\right)
Más detalles con x→oo
lim⁡x→1−((−1)xx−sin⁡(x)+tan⁡(x))=−tan⁡(1)+1+sin⁡(1)tan⁡(1)−1+sin⁡(1)\lim_{x \to 1^-}\left(\frac{\left(-1\right) x}{x - \sin{\left(x \right)}} + \tan{\left(x \right)}\right) = \frac{- \tan{\left(1 \right)} + 1 + \sin{\left(1 \right)} \tan{\left(1 \right)}}{-1 + \sin{\left(1 \right)}}
Más detalles con x→1 a la izquierda
lim⁡x→1+((−1)xx−sin⁡(x)+tan⁡(x))=−tan⁡(1)+1+sin⁡(1)tan⁡(1)−1+sin⁡(1)\lim_{x \to 1^+}\left(\frac{\left(-1\right) x}{x - \sin{\left(x \right)}} + \tan{\left(x \right)}\right) = \frac{- \tan{\left(1 \right)} + 1 + \sin{\left(1 \right)} \tan{\left(1 \right)}}{-1 + \sin{\left(1 \right)}}
Más detalles con x→1 a la derecha
lim⁡x→−∞((−1)xx−sin⁡(x)+tan⁡(x))\lim_{x \to -\infty}\left(\frac{\left(-1\right) x}{x - \sin{\left(x \right)}} + \tan{\left(x \right)}\right)
Más detalles con x→-oo
Respuesta numérica [src] 
-136806.293377731
-136806.293377731
Gráfico
Límite de la función -x/(x-sin(x))+tan(x)
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