Sr Examen
Otras calculadoras:
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- Límite de la función:
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Límite de (2-7*x+3*x^2)/(2-5*x+2*x^2)
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Límite de (2-sqrt(x))/(3-sqrt(1+2*x))
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Límite de ((1+tan(x))/(1+sin(x)))^(1/sin(x))
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Límite de (1+x)*(-1+x^3-2*x)/(-5+x^4+4*x^2)
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Expresiones idénticas
- (-x*sqrt(uno -x^ dos)+sin(sin(x)))/x^ cuatro
- ( menos x multiplicar por raíz cuadrada de (1 menos x al cuadrado ) más seno de ( seno de (x))) dividir por x en el grado 4
- ( menos x multiplicar por raíz cuadrada de (uno menos x en el grado dos) más seno de ( seno de (x))) dividir por x en el grado cuatro
- (-x*√(1-x^2)+sin(sin(x)))/x^4
- (-x*sqrt(1-x2)+sin(sin(x)))/x4
- -x*sqrt1-x2+sinsinx/x4
- (-x*sqrt(1-x²)+sin(sin(x)))/x⁴
- (-x*sqrt(1-x en el grado 2)+sin(sin(x)))/x en el grado 4
- (-xsqrt(1-x^2)+sin(sin(x)))/x^4
- (-xsqrt(1-x2)+sin(sin(x)))/x4
- -xsqrt1-x2+sinsinx/x4
- -xsqrt1-x^2+sinsinx/x^4
- (-x*sqrt(1-x^2)+sin(sin(x))) dividir por x^4
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Expresiones semejantes
- (-x*sqrt(1-x^2)-sin(sin(x)))/x^4
- (x*sqrt(1-x^2)+sin(sin(x)))/x^4
- (-x*sqrt(1+x^2)+sin(sin(x)))/x^4
- (-x*sqrt(1-x^2)+sin(sinx))/x^4
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Expresiones con funciones
- Raíz cuadrada sqrt
- sqrt(-2+x^2+3*x)-sqrt(-3+x^2)
- sqrt(1+x+x^2)-sqrt(-1+x^2-x)
- sqrt(9+x^2+4*x)-sqrt(11+x^2-8*x)
- sqrt(-1+n+n^2)-sqrt(1+n^2-n)
- sqrt(n^2+3*n)-n
- Seno sin
- sin(19*x)/sin(3*x)
- sin(x)/(x-sin(x))
- sin(x)/log(1+2*x)
- sin(x)^2/(2*x^2)
- sin(-1+x^2)/(-1+x)
- Seno sin
- sin(19*x)/sin(3*x)
- sin(x)/(x-sin(x))
- sin(x)/log(1+2*x)
- sin(x)^2/(2*x^2)
- sin(-1+x^2)/(-1+x)
Límite de la función (-x*sqrt(1-x^2)+sin(sin(x)))/x^4
Solución
Ha introducido
[src]
/ ________ \
| / 2 |
|-x*\/ 1 - x + sin(sin(x))|
lim |----------------------------|
x->0+| 4 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{- x \sqrt{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right)$$
Limit(((-x)*sqrt(1 - x^2) + sin(sin(x)))/x^4, x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(- x \sqrt{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}\right) = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{- x \sqrt{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\frac{- x \sqrt{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- x \sqrt{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}\right)}{\frac{d}{d x} x^{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{x^{2}}{\sqrt{1 - x^{2}}} - \sqrt{1 - x^{2}} + \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{4 x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\frac{x^{2}}{\sqrt{1 - x^{2}}} - \sqrt{1 - x^{2}} + \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}\right)}{\frac{d}{d x} 4 x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{x^{3}}{- x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}} + \frac{3 x}{\sqrt{1 - x^{2}}} - \sin{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{12 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\frac{x^{3}}{- x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}} + \frac{3 x}{\sqrt{1 - x^{2}}} - \sin{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}\right)}{\frac{d}{d x} 12 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{x^{6}}{- x^{6} \sqrt{1 - x^{2}} + 3 x^{4} \sqrt{1 - x^{2}} - 3 x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}} + \frac{2 x^{4} \sqrt{1 - x^{2}}}{- x^{6} + 3 x^{4} - 3 x^{2} + 1} + \frac{x^{4}}{- x^{6} \sqrt{1 - x^{2}} + 3 x^{4} \sqrt{1 - x^{2}} - 3 x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}} + \frac{6 x^{2}}{- x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}} + 3 \sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)} - \cos^{3}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \frac{3}{\sqrt{1 - x^{2}}}}{24 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{x^{6}}{- x^{6} \sqrt{1 - x^{2}} + 3 x^{4} \sqrt{1 - x^{2}} - 3 x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}} + \frac{2 x^{4} \sqrt{1 - x^{2}}}{- x^{6} + 3 x^{4} - 3 x^{2} + 1} + \frac{x^{4}}{- x^{6} \sqrt{1 - x^{2}} + 3 x^{4} \sqrt{1 - x^{2}} - 3 x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}} + \frac{6 x^{2}}{- x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}} + 3 \sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)} - \cos^{3}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \frac{3}{\sqrt{1 - x^{2}}}\right)}{\frac{d}{d x} 24 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{x^{13}}{- 24 x^{14} \sqrt{1 - x^{2}} + 168 x^{12} \sqrt{1 - x^{2}} - 504 x^{10} \sqrt{1 - x^{2}} + 840 x^{8} \sqrt{1 - x^{2}} - 840 x^{6} \sqrt{1 - x^{2}} + 504 x^{4} \sqrt{1 - x^{2}} - 168 x^{2} \sqrt{1 - x^{2}} + 24 \sqrt{1 - x^{2}}} - \frac{x^{11} \sqrt{1 - x^{2}}}{4 \left(- x^{14} + 7 x^{12} - 21 x^{10} + 35 x^{8} - 35 x^{6} + 21 x^{4} - 7 x^{2} + 1\right)} - \frac{x^{11}}{- 6 x^{14} \sqrt{1 - x^{2}} + 42 x^{12} \sqrt{1 - x^{2}} - 126 x^{10} \sqrt{1 - x^{2}} + 210 x^{8} \sqrt{1 - x^{2}} - 210 x^{6} \sqrt{1 - x^{2}} + 126 x^{4} \sqrt{1 - x^{2}} - 42 x^{2} \sqrt{1 - x^{2}} + 6 \sqrt{1 - x^{2}}} + \frac{3 x^{9} \sqrt{1 - x^{2}}}{4 \left(- x^{14} + 7 x^{12} - 21 x^{10} + 35 x^{8} - 35 x^{6} + 21 x^{4} - 7 x^{2} + 1\right)} + \frac{x^{9} \sqrt{1 - x^{2}}}{2 \left(x^{12} - 6 x^{10} + 15 x^{8} - 20 x^{6} + 15 x^{4} - 6 x^{2} + 1\right)} + \frac{x^{9}}{- 4 x^{14} \sqrt{1 - x^{2}} + 28 x^{12} \sqrt{1 - x^{2}} - 84 x^{10} \sqrt{1 - x^{2}} + 140 x^{8} \sqrt{1 - x^{2}} - 140 x^{6} \sqrt{1 - x^{2}} + 84 x^{4} \sqrt{1 - x^{2}} - 28 x^{2} \sqrt{1 - x^{2}} + 4 \sqrt{1 - x^{2}}} - \frac{3 x^{7} \sqrt{1 - x^{2}}}{4 \left(- x^{14} + 7 x^{12} - 21 x^{10} + 35 x^{8} - 35 x^{6} + 21 x^{4} - 7 x^{2} + 1\right)} - \frac{x^{7} \sqrt{1 - x^{2}}}{x^{12} - 6 x^{10} + 15 x^{8} - 20 x^{6} + 15 x^{4} - 6 x^{2} + 1} - \frac{x^{7}}{- 6 x^{14} \sqrt{1 - x^{2}} + 42 x^{12} \sqrt{1 - x^{2}} - 126 x^{10} \sqrt{1 - x^{2}} + 210 x^{8} \sqrt{1 - x^{2}} - 210 x^{6} \sqrt{1 - x^{2}} + 126 x^{4} \sqrt{1 - x^{2}} - 42 x^{2} \sqrt{1 - x^{2}} + 6 \sqrt{1 - x^{2}}} + \frac{x^{5} \sqrt{1 - x^{2}}}{4 \left(- x^{14} + 7 x^{12} - 21 x^{10} + 35 x^{8} - 35 x^{6} + 21 x^{4} - 7 x^{2} + 1\right)} + \frac{x^{5} \sqrt{1 - x^{2}}}{2 \left(x^{12} - 6 x^{10} + 15 x^{8} - 20 x^{6} + 15 x^{4} - 6 x^{2} + 1\right)} + \frac{x^{5}}{- 24 x^{14} \sqrt{1 - x^{2}} + 168 x^{12} \sqrt{1 - x^{2}} - 504 x^{10} \sqrt{1 - x^{2}} + 840 x^{8} \sqrt{1 - x^{2}} - 840 x^{6} \sqrt{1 - x^{2}} + 504 x^{4} \sqrt{1 - x^{2}} - 168 x^{2} \sqrt{1 - x^{2}} + 24 \sqrt{1 - x^{2}}} - \frac{x^{5}}{4 \left(- x^{6} \sqrt{1 - x^{2}} + 3 x^{4} \sqrt{1 - x^{2}} - 3 x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}\right)} - \frac{x^{5}}{- 4 x^{6} \sqrt{1 - x^{2}} + 12 x^{4} \sqrt{1 - x^{2}} - 12 x^{2} \sqrt{1 - x^{2}} + 4 \sqrt{1 - x^{2}}} - \frac{x^{5}}{- 12 x^{6} \sqrt{1 - x^{2}} + 36 x^{4} \sqrt{1 - x^{2}} - 36 x^{2} \sqrt{1 - x^{2}} + 12 \sqrt{1 - x^{2}}} + \frac{5 x^{3} \sqrt{1 - x^{2}}}{6 \left(- x^{6} + 3 x^{4} - 3 x^{2} + 1\right)} + \frac{x^{3}}{6 \left(- x^{6} \sqrt{1 - x^{2}} + 3 x^{4} \sqrt{1 - x^{2}} - 3 x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}\right)} + \frac{x^{3}}{- 4 x^{6} \sqrt{1 - x^{2}} + 12 x^{4} \sqrt{1 - x^{2}} - 12 x^{2} \sqrt{1 - x^{2}} + 4 \sqrt{1 - x^{2}}} + \frac{5 x}{8 \left(- x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}\right)} - \frac{\sin^{2}{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)}}{8} + \frac{\sin{\left(x \right)} \cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{4} + \frac{\sin{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{24} + \frac{\sin{\left(\sin{\left(x \right)} \right)} \cos^{4}{\left(x \right)}}{24} + \frac{\sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{6}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{x^{13}}{- 24 x^{14} \sqrt{1 - x^{2}} + 168 x^{12} \sqrt{1 - x^{2}} - 504 x^{10} \sqrt{1 - x^{2}} + 840 x^{8} \sqrt{1 - x^{2}} - 840 x^{6} \sqrt{1 - x^{2}} + 504 x^{4} \sqrt{1 - x^{2}} - 168 x^{2} \sqrt{1 - x^{2}} + 24 \sqrt{1 - x^{2}}} - \frac{x^{11} \sqrt{1 - x^{2}}}{4 \left(- x^{14} + 7 x^{12} - 21 x^{10} + 35 x^{8} - 35 x^{6} + 21 x^{4} - 7 x^{2} + 1\right)} - \frac{x^{11}}{- 6 x^{14} \sqrt{1 - x^{2}} + 42 x^{12} \sqrt{1 - x^{2}} - 126 x^{10} \sqrt{1 - x^{2}} + 210 x^{8} \sqrt{1 - x^{2}} - 210 x^{6} \sqrt{1 - x^{2}} + 126 x^{4} \sqrt{1 - x^{2}} - 42 x^{2} \sqrt{1 - x^{2}} + 6 \sqrt{1 - x^{2}}} + \frac{3 x^{9} \sqrt{1 - x^{2}}}{4 \left(- x^{14} + 7 x^{12} - 21 x^{10} + 35 x^{8} - 35 x^{6} + 21 x^{4} - 7 x^{2} + 1\right)} + \frac{x^{9} \sqrt{1 - x^{2}}}{2 \left(x^{12} - 6 x^{10} + 15 x^{8} - 20 x^{6} + 15 x^{4} - 6 x^{2} + 1\right)} + \frac{x^{9}}{- 4 x^{14} \sqrt{1 - x^{2}} + 28 x^{12} \sqrt{1 - x^{2}} - 84 x^{10} \sqrt{1 - x^{2}} + 140 x^{8} \sqrt{1 - x^{2}} - 140 x^{6} \sqrt{1 - x^{2}} + 84 x^{4} \sqrt{1 - x^{2}} - 28 x^{2} \sqrt{1 - x^{2}} + 4 \sqrt{1 - x^{2}}} - \frac{3 x^{7} \sqrt{1 - x^{2}}}{4 \left(- x^{14} + 7 x^{12} - 21 x^{10} + 35 x^{8} - 35 x^{6} + 21 x^{4} - 7 x^{2} + 1\right)} - \frac{x^{7} \sqrt{1 - x^{2}}}{x^{12} - 6 x^{10} + 15 x^{8} - 20 x^{6} + 15 x^{4} - 6 x^{2} + 1} - \frac{x^{7}}{- 6 x^{14} \sqrt{1 - x^{2}} + 42 x^{12} \sqrt{1 - x^{2}} - 126 x^{10} \sqrt{1 - x^{2}} + 210 x^{8} \sqrt{1 - x^{2}} - 210 x^{6} \sqrt{1 - x^{2}} + 126 x^{4} \sqrt{1 - x^{2}} - 42 x^{2} \sqrt{1 - x^{2}} + 6 \sqrt{1 - x^{2}}} + \frac{x^{5} \sqrt{1 - x^{2}}}{4 \left(- x^{14} + 7 x^{12} - 21 x^{10} + 35 x^{8} - 35 x^{6} + 21 x^{4} - 7 x^{2} + 1\right)} + \frac{x^{5} \sqrt{1 - x^{2}}}{2 \left(x^{12} - 6 x^{10} + 15 x^{8} - 20 x^{6} + 15 x^{4} - 6 x^{2} + 1\right)} + \frac{x^{5}}{- 24 x^{14} \sqrt{1 - x^{2}} + 168 x^{12} \sqrt{1 - x^{2}} - 504 x^{10} \sqrt{1 - x^{2}} + 840 x^{8} \sqrt{1 - x^{2}} - 840 x^{6} \sqrt{1 - x^{2}} + 504 x^{4} \sqrt{1 - x^{2}} - 168 x^{2} \sqrt{1 - x^{2}} + 24 \sqrt{1 - x^{2}}} - \frac{x^{5}}{4 \left(- x^{6} \sqrt{1 - x^{2}} + 3 x^{4} \sqrt{1 - x^{2}} - 3 x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}\right)} - \frac{x^{5}}{- 4 x^{6} \sqrt{1 - x^{2}} + 12 x^{4} \sqrt{1 - x^{2}} - 12 x^{2} \sqrt{1 - x^{2}} + 4 \sqrt{1 - x^{2}}} - \frac{x^{5}}{- 12 x^{6} \sqrt{1 - x^{2}} + 36 x^{4} \sqrt{1 - x^{2}} - 36 x^{2} \sqrt{1 - x^{2}} + 12 \sqrt{1 - x^{2}}} + \frac{5 x^{3} \sqrt{1 - x^{2}}}{6 \left(- x^{6} + 3 x^{4} - 3 x^{2} + 1\right)} + \frac{x^{3}}{6 \left(- x^{6} \sqrt{1 - x^{2}} + 3 x^{4} \sqrt{1 - x^{2}} - 3 x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}\right)} + \frac{x^{3}}{- 4 x^{6} \sqrt{1 - x^{2}} + 12 x^{4} \sqrt{1 - x^{2}} - 12 x^{2} \sqrt{1 - x^{2}} + 4 \sqrt{1 - x^{2}}} + \frac{5 x}{8 \left(- x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}\right)} - \frac{\sin^{2}{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)}}{8} + \frac{\sin{\left(x \right)} \cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{4} + \frac{\sin{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{24} + \frac{\sin{\left(\sin{\left(x \right)} \right)} \cos^{4}{\left(x \right)}}{24} + \frac{\sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{6}\right)$$
=
$$\infty$$
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
$$\lim_{x \to 0^+}\left(- x \sqrt{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}\right) = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{- x \sqrt{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\frac{- x \sqrt{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- x \sqrt{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}\right)}{\frac{d}{d x} x^{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{x^{2}}{\sqrt{1 - x^{2}}} - \sqrt{1 - x^{2}} + \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{4 x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\frac{x^{2}}{\sqrt{1 - x^{2}}} - \sqrt{1 - x^{2}} + \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}\right)}{\frac{d}{d x} 4 x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{x^{3}}{- x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}} + \frac{3 x}{\sqrt{1 - x^{2}}} - \sin{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{12 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\frac{x^{3}}{- x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}} + \frac{3 x}{\sqrt{1 - x^{2}}} - \sin{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}\right)}{\frac{d}{d x} 12 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{x^{6}}{- x^{6} \sqrt{1 - x^{2}} + 3 x^{4} \sqrt{1 - x^{2}} - 3 x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}} + \frac{2 x^{4} \sqrt{1 - x^{2}}}{- x^{6} + 3 x^{4} - 3 x^{2} + 1} + \frac{x^{4}}{- x^{6} \sqrt{1 - x^{2}} + 3 x^{4} \sqrt{1 - x^{2}} - 3 x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}} + \frac{6 x^{2}}{- x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}} + 3 \sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)} - \cos^{3}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \frac{3}{\sqrt{1 - x^{2}}}}{24 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{x^{6}}{- x^{6} \sqrt{1 - x^{2}} + 3 x^{4} \sqrt{1 - x^{2}} - 3 x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}} + \frac{2 x^{4} \sqrt{1 - x^{2}}}{- x^{6} + 3 x^{4} - 3 x^{2} + 1} + \frac{x^{4}}{- x^{6} \sqrt{1 - x^{2}} + 3 x^{4} \sqrt{1 - x^{2}} - 3 x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}} + \frac{6 x^{2}}{- x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}} + 3 \sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)} - \cos^{3}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \frac{3}{\sqrt{1 - x^{2}}}\right)}{\frac{d}{d x} 24 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{x^{13}}{- 24 x^{14} \sqrt{1 - x^{2}} + 168 x^{12} \sqrt{1 - x^{2}} - 504 x^{10} \sqrt{1 - x^{2}} + 840 x^{8} \sqrt{1 - x^{2}} - 840 x^{6} \sqrt{1 - x^{2}} + 504 x^{4} \sqrt{1 - x^{2}} - 168 x^{2} \sqrt{1 - x^{2}} + 24 \sqrt{1 - x^{2}}} - \frac{x^{11} \sqrt{1 - x^{2}}}{4 \left(- x^{14} + 7 x^{12} - 21 x^{10} + 35 x^{8} - 35 x^{6} + 21 x^{4} - 7 x^{2} + 1\right)} - \frac{x^{11}}{- 6 x^{14} \sqrt{1 - x^{2}} + 42 x^{12} \sqrt{1 - x^{2}} - 126 x^{10} \sqrt{1 - x^{2}} + 210 x^{8} \sqrt{1 - x^{2}} - 210 x^{6} \sqrt{1 - x^{2}} + 126 x^{4} \sqrt{1 - x^{2}} - 42 x^{2} \sqrt{1 - x^{2}} + 6 \sqrt{1 - x^{2}}} + \frac{3 x^{9} \sqrt{1 - x^{2}}}{4 \left(- x^{14} + 7 x^{12} - 21 x^{10} + 35 x^{8} - 35 x^{6} + 21 x^{4} - 7 x^{2} + 1\right)} + \frac{x^{9} \sqrt{1 - x^{2}}}{2 \left(x^{12} - 6 x^{10} + 15 x^{8} - 20 x^{6} + 15 x^{4} - 6 x^{2} + 1\right)} + \frac{x^{9}}{- 4 x^{14} \sqrt{1 - x^{2}} + 28 x^{12} \sqrt{1 - x^{2}} - 84 x^{10} \sqrt{1 - x^{2}} + 140 x^{8} \sqrt{1 - x^{2}} - 140 x^{6} \sqrt{1 - x^{2}} + 84 x^{4} \sqrt{1 - x^{2}} - 28 x^{2} \sqrt{1 - x^{2}} + 4 \sqrt{1 - x^{2}}} - \frac{3 x^{7} \sqrt{1 - x^{2}}}{4 \left(- x^{14} + 7 x^{12} - 21 x^{10} + 35 x^{8} - 35 x^{6} + 21 x^{4} - 7 x^{2} + 1\right)} - \frac{x^{7} \sqrt{1 - x^{2}}}{x^{12} - 6 x^{10} + 15 x^{8} - 20 x^{6} + 15 x^{4} - 6 x^{2} + 1} - \frac{x^{7}}{- 6 x^{14} \sqrt{1 - x^{2}} + 42 x^{12} \sqrt{1 - x^{2}} - 126 x^{10} \sqrt{1 - x^{2}} + 210 x^{8} \sqrt{1 - x^{2}} - 210 x^{6} \sqrt{1 - x^{2}} + 126 x^{4} \sqrt{1 - x^{2}} - 42 x^{2} \sqrt{1 - x^{2}} + 6 \sqrt{1 - x^{2}}} + \frac{x^{5} \sqrt{1 - x^{2}}}{4 \left(- x^{14} + 7 x^{12} - 21 x^{10} + 35 x^{8} - 35 x^{6} + 21 x^{4} - 7 x^{2} + 1\right)} + \frac{x^{5} \sqrt{1 - x^{2}}}{2 \left(x^{12} - 6 x^{10} + 15 x^{8} - 20 x^{6} + 15 x^{4} - 6 x^{2} + 1\right)} + \frac{x^{5}}{- 24 x^{14} \sqrt{1 - x^{2}} + 168 x^{12} \sqrt{1 - x^{2}} - 504 x^{10} \sqrt{1 - x^{2}} + 840 x^{8} \sqrt{1 - x^{2}} - 840 x^{6} \sqrt{1 - x^{2}} + 504 x^{4} \sqrt{1 - x^{2}} - 168 x^{2} \sqrt{1 - x^{2}} + 24 \sqrt{1 - x^{2}}} - \frac{x^{5}}{4 \left(- x^{6} \sqrt{1 - x^{2}} + 3 x^{4} \sqrt{1 - x^{2}} - 3 x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}\right)} - \frac{x^{5}}{- 4 x^{6} \sqrt{1 - x^{2}} + 12 x^{4} \sqrt{1 - x^{2}} - 12 x^{2} \sqrt{1 - x^{2}} + 4 \sqrt{1 - x^{2}}} - \frac{x^{5}}{- 12 x^{6} \sqrt{1 - x^{2}} + 36 x^{4} \sqrt{1 - x^{2}} - 36 x^{2} \sqrt{1 - x^{2}} + 12 \sqrt{1 - x^{2}}} + \frac{5 x^{3} \sqrt{1 - x^{2}}}{6 \left(- x^{6} + 3 x^{4} - 3 x^{2} + 1\right)} + \frac{x^{3}}{6 \left(- x^{6} \sqrt{1 - x^{2}} + 3 x^{4} \sqrt{1 - x^{2}} - 3 x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}\right)} + \frac{x^{3}}{- 4 x^{6} \sqrt{1 - x^{2}} + 12 x^{4} \sqrt{1 - x^{2}} - 12 x^{2} \sqrt{1 - x^{2}} + 4 \sqrt{1 - x^{2}}} + \frac{5 x}{8 \left(- x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}\right)} - \frac{\sin^{2}{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)}}{8} + \frac{\sin{\left(x \right)} \cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{4} + \frac{\sin{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{24} + \frac{\sin{\left(\sin{\left(x \right)} \right)} \cos^{4}{\left(x \right)}}{24} + \frac{\sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{6}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{x^{13}}{- 24 x^{14} \sqrt{1 - x^{2}} + 168 x^{12} \sqrt{1 - x^{2}} - 504 x^{10} \sqrt{1 - x^{2}} + 840 x^{8} \sqrt{1 - x^{2}} - 840 x^{6} \sqrt{1 - x^{2}} + 504 x^{4} \sqrt{1 - x^{2}} - 168 x^{2} \sqrt{1 - x^{2}} + 24 \sqrt{1 - x^{2}}} - \frac{x^{11} \sqrt{1 - x^{2}}}{4 \left(- x^{14} + 7 x^{12} - 21 x^{10} + 35 x^{8} - 35 x^{6} + 21 x^{4} - 7 x^{2} + 1\right)} - \frac{x^{11}}{- 6 x^{14} \sqrt{1 - x^{2}} + 42 x^{12} \sqrt{1 - x^{2}} - 126 x^{10} \sqrt{1 - x^{2}} + 210 x^{8} \sqrt{1 - x^{2}} - 210 x^{6} \sqrt{1 - x^{2}} + 126 x^{4} \sqrt{1 - x^{2}} - 42 x^{2} \sqrt{1 - x^{2}} + 6 \sqrt{1 - x^{2}}} + \frac{3 x^{9} \sqrt{1 - x^{2}}}{4 \left(- x^{14} + 7 x^{12} - 21 x^{10} + 35 x^{8} - 35 x^{6} + 21 x^{4} - 7 x^{2} + 1\right)} + \frac{x^{9} \sqrt{1 - x^{2}}}{2 \left(x^{12} - 6 x^{10} + 15 x^{8} - 20 x^{6} + 15 x^{4} - 6 x^{2} + 1\right)} + \frac{x^{9}}{- 4 x^{14} \sqrt{1 - x^{2}} + 28 x^{12} \sqrt{1 - x^{2}} - 84 x^{10} \sqrt{1 - x^{2}} + 140 x^{8} \sqrt{1 - x^{2}} - 140 x^{6} \sqrt{1 - x^{2}} + 84 x^{4} \sqrt{1 - x^{2}} - 28 x^{2} \sqrt{1 - x^{2}} + 4 \sqrt{1 - x^{2}}} - \frac{3 x^{7} \sqrt{1 - x^{2}}}{4 \left(- x^{14} + 7 x^{12} - 21 x^{10} + 35 x^{8} - 35 x^{6} + 21 x^{4} - 7 x^{2} + 1\right)} - \frac{x^{7} \sqrt{1 - x^{2}}}{x^{12} - 6 x^{10} + 15 x^{8} - 20 x^{6} + 15 x^{4} - 6 x^{2} + 1} - \frac{x^{7}}{- 6 x^{14} \sqrt{1 - x^{2}} + 42 x^{12} \sqrt{1 - x^{2}} - 126 x^{10} \sqrt{1 - x^{2}} + 210 x^{8} \sqrt{1 - x^{2}} - 210 x^{6} \sqrt{1 - x^{2}} + 126 x^{4} \sqrt{1 - x^{2}} - 42 x^{2} \sqrt{1 - x^{2}} + 6 \sqrt{1 - x^{2}}} + \frac{x^{5} \sqrt{1 - x^{2}}}{4 \left(- x^{14} + 7 x^{12} - 21 x^{10} + 35 x^{8} - 35 x^{6} + 21 x^{4} - 7 x^{2} + 1\right)} + \frac{x^{5} \sqrt{1 - x^{2}}}{2 \left(x^{12} - 6 x^{10} + 15 x^{8} - 20 x^{6} + 15 x^{4} - 6 x^{2} + 1\right)} + \frac{x^{5}}{- 24 x^{14} \sqrt{1 - x^{2}} + 168 x^{12} \sqrt{1 - x^{2}} - 504 x^{10} \sqrt{1 - x^{2}} + 840 x^{8} \sqrt{1 - x^{2}} - 840 x^{6} \sqrt{1 - x^{2}} + 504 x^{4} \sqrt{1 - x^{2}} - 168 x^{2} \sqrt{1 - x^{2}} + 24 \sqrt{1 - x^{2}}} - \frac{x^{5}}{4 \left(- x^{6} \sqrt{1 - x^{2}} + 3 x^{4} \sqrt{1 - x^{2}} - 3 x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}\right)} - \frac{x^{5}}{- 4 x^{6} \sqrt{1 - x^{2}} + 12 x^{4} \sqrt{1 - x^{2}} - 12 x^{2} \sqrt{1 - x^{2}} + 4 \sqrt{1 - x^{2}}} - \frac{x^{5}}{- 12 x^{6} \sqrt{1 - x^{2}} + 36 x^{4} \sqrt{1 - x^{2}} - 36 x^{2} \sqrt{1 - x^{2}} + 12 \sqrt{1 - x^{2}}} + \frac{5 x^{3} \sqrt{1 - x^{2}}}{6 \left(- x^{6} + 3 x^{4} - 3 x^{2} + 1\right)} + \frac{x^{3}}{6 \left(- x^{6} \sqrt{1 - x^{2}} + 3 x^{4} \sqrt{1 - x^{2}} - 3 x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}\right)} + \frac{x^{3}}{- 4 x^{6} \sqrt{1 - x^{2}} + 12 x^{4} \sqrt{1 - x^{2}} - 12 x^{2} \sqrt{1 - x^{2}} + 4 \sqrt{1 - x^{2}}} + \frac{5 x}{8 \left(- x^{2} \sqrt{1 - x^{2}} + \sqrt{1 - x^{2}}\right)} - \frac{\sin^{2}{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)}}{8} + \frac{\sin{\left(x \right)} \cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{4} + \frac{\sin{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{24} + \frac{\sin{\left(\sin{\left(x \right)} \right)} \cos^{4}{\left(x \right)}}{24} + \frac{\sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{6}\right)$$
=
$$\infty$$
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]
/ ________ \
| / 2 |
|-x*\/ 1 - x + sin(sin(x))|
lim |----------------------------|
x->0+| 4 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{- x \sqrt{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right)$$
oo
$$\infty$$
= 25.168156743669
/ ________ \
| / 2 |
|-x*\/ 1 - x + sin(sin(x))|
lim |----------------------------|
x->0-| 4 |
\ x /
$$\lim_{x \to 0^-}\left(\frac{- x \sqrt{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right)$$
-oo
$$-\infty$$
= -25.168156743669
= -25.168156743669
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{- x \sqrt{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right) = \infty$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{- x \sqrt{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{- x \sqrt{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right) = 0$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{- x \sqrt{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right) = \sin{\left(\sin{\left(1 \right)} \right)}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{- x \sqrt{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right) = \sin{\left(\sin{\left(1 \right)} \right)}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{- x \sqrt{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right) = 0$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{- x \sqrt{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{- x \sqrt{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right) = 0$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{- x \sqrt{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right) = \sin{\left(\sin{\left(1 \right)} \right)}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{- x \sqrt{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right) = \sin{\left(\sin{\left(1 \right)} \right)}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{- x \sqrt{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4}}\right) = 0$$
Más detalles con x→-oo