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- Límite de la función:
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Límite de (-9+x^2)/(3+x)
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Límite de x^2/(1-cos(6*x))
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Límite de (4+x^2-5*x)/(8+x^2-6*x)
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Límite de (-3+x^2-2*x)/(-9+x^2)
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Expresiones idénticas
- (-x*(uno -x^ dos)^(uno / tres)+sin(sin(x)))/(x^ cuatro *(- uno +x))
- ( menos x multiplicar por (1 menos x al cuadrado ) en el grado (1 dividir por 3) más seno de ( seno de (x))) dividir por (x en el grado 4 multiplicar por ( menos 1 más x))
- ( menos x multiplicar por (uno menos x en el grado dos) en el grado (uno dividir por tres) más seno de ( seno de (x))) dividir por (x en el grado cuatro multiplicar por ( menos uno más x))
- (-x*(1-x2)(1/3)+sin(sin(x)))/(x4*(-1+x))
- -x*1-x21/3+sinsinx/x4*-1+x
- (-x*(1-x²)^(1/3)+sin(sin(x)))/(x⁴*(-1+x))
- (-x*(1-x en el grado 2) en el grado (1/3)+sin(sin(x)))/(x en el grado 4*(-1+x))
- (-x(1-x^2)^(1/3)+sin(sin(x)))/(x^4(-1+x))
- (-x(1-x2)(1/3)+sin(sin(x)))/(x4(-1+x))
- -x1-x21/3+sinsinx/x4-1+x
- -x1-x^2^1/3+sinsinx/x^4-1+x
- (-x*(1-x^2)^(1 dividir por 3)+sin(sin(x))) dividir por (x^4*(-1+x))
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Expresiones semejantes
- (-x*(1+x^2)^(1/3)+sin(sin(x)))/(x^4*(-1+x))
- (x*(1-x^2)^(1/3)+sin(sin(x)))/(x^4*(-1+x))
- (-x*(1-x^2)^(1/3)+sin(sin(x)))/(x^4*(1+x))
- (-x*(1-x^2)^(1/3)-sin(sin(x)))/(x^4*(-1+x))
- (-x*(1-x^2)^(1/3)+sin(sin(x)))/(x^4*(-1-x))
- (-x*(1-x^2)^(1/3)+sin(sinx))/(x^4*(-1+x))
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Expresiones con funciones
- Seno sin
- sin(5*x)/sin(2*x)
- sin(3*x)/(3*x)
- sin(x)/|x|
- sin(x)/asin(x)
- sin(k*x)/x
- Seno sin
- sin(5*x)/sin(2*x)
- sin(3*x)/(3*x)
- sin(x)/|x|
- sin(x)/asin(x)
- sin(k*x)/x
Límite de la función (-x*(1-x^2)^(1/3)+sin(sin(x)))/(x^4*(-1+x))
Solución
Ha introducido
[src]
/ ________ \
| 3 / 2 |
|-x*\/ 1 - x + sin(sin(x))|
lim |----------------------------|
x->0+| 4 |
\ x *(-1 + x) /
$$\lim_{x \to 0^+}\left(\frac{- x \sqrt[3]{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4} \left(x - 1\right)}\right)$$
Limit(((-x)*(1 - x^2)^(1/3) + sin(sin(x)))/((x^4*(-1 + x))), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(- x \sqrt[3]{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+}\left(x^{5} - x^{4}\right) = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{- x \sqrt[3]{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4} \left(x - 1\right)}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\frac{- x \sqrt[3]{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4} \left(x - 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- x \sqrt[3]{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}\right)}{\frac{d}{d x} \left(x^{5} - x^{4}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{2 x^{2}}{3 \left(1 - x^{2}\right)^{\frac{2}{3}}} - \sqrt[3]{1 - x^{2}} + \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{5 x^{4} - 4 x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\frac{2 x^{2}}{3 \left(1 - x^{2}\right)^{\frac{2}{3}}} - \sqrt[3]{1 - x^{2}} + \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}\right)}{\frac{d}{d x} \left(5 x^{4} - 4 x^{3}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{8 x^{3}}{9 \left(- x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} + \frac{2 x}{\left(1 - x^{2}\right)^{\frac{2}{3}}} - \sin{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{20 x^{3} - 12 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\frac{8 x^{3}}{9 \left(- x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} + \frac{2 x}{\left(1 - x^{2}\right)^{\frac{2}{3}}} - \sin{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}\right)}{\frac{d}{d x} \left(20 x^{3} - 12 x^{2}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{32 x^{6}}{- 27 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 81 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 81 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 27 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{16 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}}}{9 \left(- x^{6} \sqrt[3]{1 - x^{2}} + 3 x^{4} \sqrt[3]{1 - x^{2}} - 3 x^{2} \sqrt[3]{1 - x^{2}} + \sqrt[3]{1 - x^{2}}\right)} + \frac{32 x^{4}}{- 27 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 81 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 81 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 27 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{16 x^{2}}{3 \left(- x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} + 3 \sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)} - \cos^{3}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \frac{2}{\left(1 - x^{2}\right)^{\frac{2}{3}}}}{60 x^{2} - 24 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{32 x^{6}}{- 27 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 81 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 81 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 27 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{16 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}}}{9 \left(- x^{6} \sqrt[3]{1 - x^{2}} + 3 x^{4} \sqrt[3]{1 - x^{2}} - 3 x^{2} \sqrt[3]{1 - x^{2}} + \sqrt[3]{1 - x^{2}}\right)} + \frac{32 x^{4}}{- 27 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 81 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 81 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 27 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{16 x^{2}}{3 \left(- x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} + 3 \sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)} - \cos^{3}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \frac{2}{\left(1 - x^{2}\right)^{\frac{2}{3}}}\right)}{\frac{d}{d x} \left(60 x^{2} - 24 x\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{1152 x^{13}}{- 729 x^{14} \left(1 - x^{2}\right)^{\frac{2}{3}} + 5103 x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 15309 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 25515 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 25515 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15309 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 5103 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 729 \left(1 - x^{2}\right)^{\frac{2}{3}}} - \frac{5184 x^{11} \left(1 - x^{2}\right)^{\frac{2}{3}}}{- 729 x^{14} \sqrt[3]{1 - x^{2}} + 5103 x^{12} \sqrt[3]{1 - x^{2}} - 15309 x^{10} \sqrt[3]{1 - x^{2}} + 25515 x^{8} \sqrt[3]{1 - x^{2}} - 25515 x^{6} \sqrt[3]{1 - x^{2}} + 15309 x^{4} \sqrt[3]{1 - x^{2}} - 5103 x^{2} \sqrt[3]{1 - x^{2}} + 729 \sqrt[3]{1 - x^{2}}} - \frac{4608 x^{11}}{- 729 x^{14} \left(1 - x^{2}\right)^{\frac{2}{3}} + 5103 x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 15309 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 25515 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 25515 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15309 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 5103 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 729 \left(1 - x^{2}\right)^{\frac{2}{3}}} - \frac{320 x^{11}}{27 \left(x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 20 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} + \frac{15552 x^{9} \left(1 - x^{2}\right)^{\frac{2}{3}}}{- 729 x^{14} \sqrt[3]{1 - x^{2}} + 5103 x^{12} \sqrt[3]{1 - x^{2}} - 15309 x^{10} \sqrt[3]{1 - x^{2}} + 25515 x^{8} \sqrt[3]{1 - x^{2}} - 25515 x^{6} \sqrt[3]{1 - x^{2}} + 15309 x^{4} \sqrt[3]{1 - x^{2}} - 5103 x^{2} \sqrt[3]{1 - x^{2}} + 729 \sqrt[3]{1 - x^{2}}} + \frac{6912 x^{9}}{- 729 x^{14} \left(1 - x^{2}\right)^{\frac{2}{3}} + 5103 x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 15309 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 25515 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 25515 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15309 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 5103 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 729 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{320 x^{9}}{9 \left(x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 20 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} - \frac{15552 x^{7} \left(1 - x^{2}\right)^{\frac{2}{3}}}{- 729 x^{14} \sqrt[3]{1 - x^{2}} + 5103 x^{12} \sqrt[3]{1 - x^{2}} - 15309 x^{10} \sqrt[3]{1 - x^{2}} + 25515 x^{8} \sqrt[3]{1 - x^{2}} - 25515 x^{6} \sqrt[3]{1 - x^{2}} + 15309 x^{4} \sqrt[3]{1 - x^{2}} - 5103 x^{2} \sqrt[3]{1 - x^{2}} + 729 \sqrt[3]{1 - x^{2}}} - \frac{4608 x^{7}}{- 729 x^{14} \left(1 - x^{2}\right)^{\frac{2}{3}} + 5103 x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 15309 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 25515 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 25515 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15309 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 5103 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 729 \left(1 - x^{2}\right)^{\frac{2}{3}}} - \frac{320 x^{7}}{9 \left(x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 20 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} + \frac{5184 x^{5} \left(1 - x^{2}\right)^{\frac{2}{3}}}{- 729 x^{14} \sqrt[3]{1 - x^{2}} + 5103 x^{12} \sqrt[3]{1 - x^{2}} - 15309 x^{10} \sqrt[3]{1 - x^{2}} + 25515 x^{8} \sqrt[3]{1 - x^{2}} - 25515 x^{6} \sqrt[3]{1 - x^{2}} + 15309 x^{4} \sqrt[3]{1 - x^{2}} - 5103 x^{2} \sqrt[3]{1 - x^{2}} + 729 \sqrt[3]{1 - x^{2}}} + \frac{1152 x^{5}}{- 729 x^{14} \left(1 - x^{2}\right)^{\frac{2}{3}} + 5103 x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 15309 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 25515 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 25515 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15309 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 5103 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 729 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{320 x^{5}}{27 \left(x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 20 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} - \frac{64 x^{5}}{- 9 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 27 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 27 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 9 \left(1 - x^{2}\right)^{\frac{2}{3}}} - \frac{256 x^{5}}{- 27 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 81 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 81 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 27 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{160 x^{3} \left(1 - x^{2}\right)^{\frac{2}{3}}}{9 \left(- x^{6} \sqrt[3]{1 - x^{2}} + 3 x^{4} \sqrt[3]{1 - x^{2}} - 3 x^{2} \sqrt[3]{1 - x^{2}} + \sqrt[3]{1 - x^{2}}\right)} + \frac{64 x^{3}}{- 9 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 27 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 27 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 9 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{128 x^{3}}{- 27 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 81 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 81 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 27 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{40 x}{3 \left(- x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} - 3 \sin^{2}{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} + 6 \sin{\left(x \right)} \cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \sin{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \sin{\left(\sin{\left(x \right)} \right)} \cos^{4}{\left(x \right)} + 4 \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{120 x - 24}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{1152 x^{13}}{- 729 x^{14} \left(1 - x^{2}\right)^{\frac{2}{3}} + 5103 x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 15309 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 25515 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 25515 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15309 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 5103 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 729 \left(1 - x^{2}\right)^{\frac{2}{3}}} - \frac{5184 x^{11} \left(1 - x^{2}\right)^{\frac{2}{3}}}{- 729 x^{14} \sqrt[3]{1 - x^{2}} + 5103 x^{12} \sqrt[3]{1 - x^{2}} - 15309 x^{10} \sqrt[3]{1 - x^{2}} + 25515 x^{8} \sqrt[3]{1 - x^{2}} - 25515 x^{6} \sqrt[3]{1 - x^{2}} + 15309 x^{4} \sqrt[3]{1 - x^{2}} - 5103 x^{2} \sqrt[3]{1 - x^{2}} + 729 \sqrt[3]{1 - x^{2}}} - \frac{4608 x^{11}}{- 729 x^{14} \left(1 - x^{2}\right)^{\frac{2}{3}} + 5103 x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 15309 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 25515 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 25515 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15309 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 5103 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 729 \left(1 - x^{2}\right)^{\frac{2}{3}}} - \frac{320 x^{11}}{27 \left(x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 20 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} + \frac{15552 x^{9} \left(1 - x^{2}\right)^{\frac{2}{3}}}{- 729 x^{14} \sqrt[3]{1 - x^{2}} + 5103 x^{12} \sqrt[3]{1 - x^{2}} - 15309 x^{10} \sqrt[3]{1 - x^{2}} + 25515 x^{8} \sqrt[3]{1 - x^{2}} - 25515 x^{6} \sqrt[3]{1 - x^{2}} + 15309 x^{4} \sqrt[3]{1 - x^{2}} - 5103 x^{2} \sqrt[3]{1 - x^{2}} + 729 \sqrt[3]{1 - x^{2}}} + \frac{6912 x^{9}}{- 729 x^{14} \left(1 - x^{2}\right)^{\frac{2}{3}} + 5103 x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 15309 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 25515 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 25515 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15309 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 5103 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 729 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{320 x^{9}}{9 \left(x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 20 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} - \frac{15552 x^{7} \left(1 - x^{2}\right)^{\frac{2}{3}}}{- 729 x^{14} \sqrt[3]{1 - x^{2}} + 5103 x^{12} \sqrt[3]{1 - x^{2}} - 15309 x^{10} \sqrt[3]{1 - x^{2}} + 25515 x^{8} \sqrt[3]{1 - x^{2}} - 25515 x^{6} \sqrt[3]{1 - x^{2}} + 15309 x^{4} \sqrt[3]{1 - x^{2}} - 5103 x^{2} \sqrt[3]{1 - x^{2}} + 729 \sqrt[3]{1 - x^{2}}} - \frac{4608 x^{7}}{- 729 x^{14} \left(1 - x^{2}\right)^{\frac{2}{3}} + 5103 x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 15309 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 25515 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 25515 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15309 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 5103 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 729 \left(1 - x^{2}\right)^{\frac{2}{3}}} - \frac{320 x^{7}}{9 \left(x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 20 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} + \frac{5184 x^{5} \left(1 - x^{2}\right)^{\frac{2}{3}}}{- 729 x^{14} \sqrt[3]{1 - x^{2}} + 5103 x^{12} \sqrt[3]{1 - x^{2}} - 15309 x^{10} \sqrt[3]{1 - x^{2}} + 25515 x^{8} \sqrt[3]{1 - x^{2}} - 25515 x^{6} \sqrt[3]{1 - x^{2}} + 15309 x^{4} \sqrt[3]{1 - x^{2}} - 5103 x^{2} \sqrt[3]{1 - x^{2}} + 729 \sqrt[3]{1 - x^{2}}} + \frac{1152 x^{5}}{- 729 x^{14} \left(1 - x^{2}\right)^{\frac{2}{3}} + 5103 x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 15309 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 25515 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 25515 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15309 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 5103 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 729 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{320 x^{5}}{27 \left(x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 20 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} - \frac{64 x^{5}}{- 9 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 27 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 27 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 9 \left(1 - x^{2}\right)^{\frac{2}{3}}} - \frac{256 x^{5}}{- 27 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 81 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 81 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 27 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{160 x^{3} \left(1 - x^{2}\right)^{\frac{2}{3}}}{9 \left(- x^{6} \sqrt[3]{1 - x^{2}} + 3 x^{4} \sqrt[3]{1 - x^{2}} - 3 x^{2} \sqrt[3]{1 - x^{2}} + \sqrt[3]{1 - x^{2}}\right)} + \frac{64 x^{3}}{- 9 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 27 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 27 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 9 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{128 x^{3}}{- 27 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 81 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 81 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 27 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{40 x}{3 \left(- x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} - 3 \sin^{2}{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} + 6 \sin{\left(x \right)} \cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \sin{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \sin{\left(\sin{\left(x \right)} \right)} \cos^{4}{\left(x \right)} + 4 \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{120 x - 24}\right)$$
=
$$0$$
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
$$\lim_{x \to 0^+}\left(- x \sqrt[3]{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+}\left(x^{5} - x^{4}\right) = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{- x \sqrt[3]{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4} \left(x - 1\right)}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\frac{- x \sqrt[3]{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4} \left(x - 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- x \sqrt[3]{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}\right)}{\frac{d}{d x} \left(x^{5} - x^{4}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{2 x^{2}}{3 \left(1 - x^{2}\right)^{\frac{2}{3}}} - \sqrt[3]{1 - x^{2}} + \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{5 x^{4} - 4 x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\frac{2 x^{2}}{3 \left(1 - x^{2}\right)^{\frac{2}{3}}} - \sqrt[3]{1 - x^{2}} + \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}\right)}{\frac{d}{d x} \left(5 x^{4} - 4 x^{3}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{8 x^{3}}{9 \left(- x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} + \frac{2 x}{\left(1 - x^{2}\right)^{\frac{2}{3}}} - \sin{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{20 x^{3} - 12 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\frac{8 x^{3}}{9 \left(- x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} + \frac{2 x}{\left(1 - x^{2}\right)^{\frac{2}{3}}} - \sin{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}\right)}{\frac{d}{d x} \left(20 x^{3} - 12 x^{2}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{32 x^{6}}{- 27 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 81 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 81 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 27 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{16 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}}}{9 \left(- x^{6} \sqrt[3]{1 - x^{2}} + 3 x^{4} \sqrt[3]{1 - x^{2}} - 3 x^{2} \sqrt[3]{1 - x^{2}} + \sqrt[3]{1 - x^{2}}\right)} + \frac{32 x^{4}}{- 27 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 81 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 81 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 27 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{16 x^{2}}{3 \left(- x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} + 3 \sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)} - \cos^{3}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \frac{2}{\left(1 - x^{2}\right)^{\frac{2}{3}}}}{60 x^{2} - 24 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{32 x^{6}}{- 27 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 81 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 81 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 27 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{16 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}}}{9 \left(- x^{6} \sqrt[3]{1 - x^{2}} + 3 x^{4} \sqrt[3]{1 - x^{2}} - 3 x^{2} \sqrt[3]{1 - x^{2}} + \sqrt[3]{1 - x^{2}}\right)} + \frac{32 x^{4}}{- 27 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 81 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 81 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 27 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{16 x^{2}}{3 \left(- x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} + 3 \sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)} - \cos^{3}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \frac{2}{\left(1 - x^{2}\right)^{\frac{2}{3}}}\right)}{\frac{d}{d x} \left(60 x^{2} - 24 x\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{1152 x^{13}}{- 729 x^{14} \left(1 - x^{2}\right)^{\frac{2}{3}} + 5103 x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 15309 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 25515 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 25515 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15309 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 5103 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 729 \left(1 - x^{2}\right)^{\frac{2}{3}}} - \frac{5184 x^{11} \left(1 - x^{2}\right)^{\frac{2}{3}}}{- 729 x^{14} \sqrt[3]{1 - x^{2}} + 5103 x^{12} \sqrt[3]{1 - x^{2}} - 15309 x^{10} \sqrt[3]{1 - x^{2}} + 25515 x^{8} \sqrt[3]{1 - x^{2}} - 25515 x^{6} \sqrt[3]{1 - x^{2}} + 15309 x^{4} \sqrt[3]{1 - x^{2}} - 5103 x^{2} \sqrt[3]{1 - x^{2}} + 729 \sqrt[3]{1 - x^{2}}} - \frac{4608 x^{11}}{- 729 x^{14} \left(1 - x^{2}\right)^{\frac{2}{3}} + 5103 x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 15309 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 25515 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 25515 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15309 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 5103 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 729 \left(1 - x^{2}\right)^{\frac{2}{3}}} - \frac{320 x^{11}}{27 \left(x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 20 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} + \frac{15552 x^{9} \left(1 - x^{2}\right)^{\frac{2}{3}}}{- 729 x^{14} \sqrt[3]{1 - x^{2}} + 5103 x^{12} \sqrt[3]{1 - x^{2}} - 15309 x^{10} \sqrt[3]{1 - x^{2}} + 25515 x^{8} \sqrt[3]{1 - x^{2}} - 25515 x^{6} \sqrt[3]{1 - x^{2}} + 15309 x^{4} \sqrt[3]{1 - x^{2}} - 5103 x^{2} \sqrt[3]{1 - x^{2}} + 729 \sqrt[3]{1 - x^{2}}} + \frac{6912 x^{9}}{- 729 x^{14} \left(1 - x^{2}\right)^{\frac{2}{3}} + 5103 x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 15309 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 25515 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 25515 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15309 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 5103 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 729 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{320 x^{9}}{9 \left(x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 20 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} - \frac{15552 x^{7} \left(1 - x^{2}\right)^{\frac{2}{3}}}{- 729 x^{14} \sqrt[3]{1 - x^{2}} + 5103 x^{12} \sqrt[3]{1 - x^{2}} - 15309 x^{10} \sqrt[3]{1 - x^{2}} + 25515 x^{8} \sqrt[3]{1 - x^{2}} - 25515 x^{6} \sqrt[3]{1 - x^{2}} + 15309 x^{4} \sqrt[3]{1 - x^{2}} - 5103 x^{2} \sqrt[3]{1 - x^{2}} + 729 \sqrt[3]{1 - x^{2}}} - \frac{4608 x^{7}}{- 729 x^{14} \left(1 - x^{2}\right)^{\frac{2}{3}} + 5103 x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 15309 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 25515 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 25515 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15309 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 5103 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 729 \left(1 - x^{2}\right)^{\frac{2}{3}}} - \frac{320 x^{7}}{9 \left(x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 20 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} + \frac{5184 x^{5} \left(1 - x^{2}\right)^{\frac{2}{3}}}{- 729 x^{14} \sqrt[3]{1 - x^{2}} + 5103 x^{12} \sqrt[3]{1 - x^{2}} - 15309 x^{10} \sqrt[3]{1 - x^{2}} + 25515 x^{8} \sqrt[3]{1 - x^{2}} - 25515 x^{6} \sqrt[3]{1 - x^{2}} + 15309 x^{4} \sqrt[3]{1 - x^{2}} - 5103 x^{2} \sqrt[3]{1 - x^{2}} + 729 \sqrt[3]{1 - x^{2}}} + \frac{1152 x^{5}}{- 729 x^{14} \left(1 - x^{2}\right)^{\frac{2}{3}} + 5103 x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 15309 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 25515 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 25515 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15309 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 5103 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 729 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{320 x^{5}}{27 \left(x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 20 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} - \frac{64 x^{5}}{- 9 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 27 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 27 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 9 \left(1 - x^{2}\right)^{\frac{2}{3}}} - \frac{256 x^{5}}{- 27 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 81 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 81 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 27 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{160 x^{3} \left(1 - x^{2}\right)^{\frac{2}{3}}}{9 \left(- x^{6} \sqrt[3]{1 - x^{2}} + 3 x^{4} \sqrt[3]{1 - x^{2}} - 3 x^{2} \sqrt[3]{1 - x^{2}} + \sqrt[3]{1 - x^{2}}\right)} + \frac{64 x^{3}}{- 9 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 27 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 27 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 9 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{128 x^{3}}{- 27 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 81 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 81 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 27 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{40 x}{3 \left(- x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} - 3 \sin^{2}{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} + 6 \sin{\left(x \right)} \cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \sin{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \sin{\left(\sin{\left(x \right)} \right)} \cos^{4}{\left(x \right)} + 4 \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{120 x - 24}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{1152 x^{13}}{- 729 x^{14} \left(1 - x^{2}\right)^{\frac{2}{3}} + 5103 x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 15309 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 25515 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 25515 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15309 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 5103 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 729 \left(1 - x^{2}\right)^{\frac{2}{3}}} - \frac{5184 x^{11} \left(1 - x^{2}\right)^{\frac{2}{3}}}{- 729 x^{14} \sqrt[3]{1 - x^{2}} + 5103 x^{12} \sqrt[3]{1 - x^{2}} - 15309 x^{10} \sqrt[3]{1 - x^{2}} + 25515 x^{8} \sqrt[3]{1 - x^{2}} - 25515 x^{6} \sqrt[3]{1 - x^{2}} + 15309 x^{4} \sqrt[3]{1 - x^{2}} - 5103 x^{2} \sqrt[3]{1 - x^{2}} + 729 \sqrt[3]{1 - x^{2}}} - \frac{4608 x^{11}}{- 729 x^{14} \left(1 - x^{2}\right)^{\frac{2}{3}} + 5103 x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 15309 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 25515 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 25515 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15309 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 5103 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 729 \left(1 - x^{2}\right)^{\frac{2}{3}}} - \frac{320 x^{11}}{27 \left(x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 20 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} + \frac{15552 x^{9} \left(1 - x^{2}\right)^{\frac{2}{3}}}{- 729 x^{14} \sqrt[3]{1 - x^{2}} + 5103 x^{12} \sqrt[3]{1 - x^{2}} - 15309 x^{10} \sqrt[3]{1 - x^{2}} + 25515 x^{8} \sqrt[3]{1 - x^{2}} - 25515 x^{6} \sqrt[3]{1 - x^{2}} + 15309 x^{4} \sqrt[3]{1 - x^{2}} - 5103 x^{2} \sqrt[3]{1 - x^{2}} + 729 \sqrt[3]{1 - x^{2}}} + \frac{6912 x^{9}}{- 729 x^{14} \left(1 - x^{2}\right)^{\frac{2}{3}} + 5103 x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 15309 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 25515 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 25515 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15309 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 5103 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 729 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{320 x^{9}}{9 \left(x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 20 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} - \frac{15552 x^{7} \left(1 - x^{2}\right)^{\frac{2}{3}}}{- 729 x^{14} \sqrt[3]{1 - x^{2}} + 5103 x^{12} \sqrt[3]{1 - x^{2}} - 15309 x^{10} \sqrt[3]{1 - x^{2}} + 25515 x^{8} \sqrt[3]{1 - x^{2}} - 25515 x^{6} \sqrt[3]{1 - x^{2}} + 15309 x^{4} \sqrt[3]{1 - x^{2}} - 5103 x^{2} \sqrt[3]{1 - x^{2}} + 729 \sqrt[3]{1 - x^{2}}} - \frac{4608 x^{7}}{- 729 x^{14} \left(1 - x^{2}\right)^{\frac{2}{3}} + 5103 x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 15309 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 25515 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 25515 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15309 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 5103 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 729 \left(1 - x^{2}\right)^{\frac{2}{3}}} - \frac{320 x^{7}}{9 \left(x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 20 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} + \frac{5184 x^{5} \left(1 - x^{2}\right)^{\frac{2}{3}}}{- 729 x^{14} \sqrt[3]{1 - x^{2}} + 5103 x^{12} \sqrt[3]{1 - x^{2}} - 15309 x^{10} \sqrt[3]{1 - x^{2}} + 25515 x^{8} \sqrt[3]{1 - x^{2}} - 25515 x^{6} \sqrt[3]{1 - x^{2}} + 15309 x^{4} \sqrt[3]{1 - x^{2}} - 5103 x^{2} \sqrt[3]{1 - x^{2}} + 729 \sqrt[3]{1 - x^{2}}} + \frac{1152 x^{5}}{- 729 x^{14} \left(1 - x^{2}\right)^{\frac{2}{3}} + 5103 x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 15309 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 25515 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 25515 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15309 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 5103 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 729 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{320 x^{5}}{27 \left(x^{12} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{10} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{8} \left(1 - x^{2}\right)^{\frac{2}{3}} - 20 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 15 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 6 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} - \frac{64 x^{5}}{- 9 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 27 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 27 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 9 \left(1 - x^{2}\right)^{\frac{2}{3}}} - \frac{256 x^{5}}{- 27 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 81 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 81 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 27 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{160 x^{3} \left(1 - x^{2}\right)^{\frac{2}{3}}}{9 \left(- x^{6} \sqrt[3]{1 - x^{2}} + 3 x^{4} \sqrt[3]{1 - x^{2}} - 3 x^{2} \sqrt[3]{1 - x^{2}} + \sqrt[3]{1 - x^{2}}\right)} + \frac{64 x^{3}}{- 9 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 27 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 27 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 9 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{128 x^{3}}{- 27 x^{6} \left(1 - x^{2}\right)^{\frac{2}{3}} + 81 x^{4} \left(1 - x^{2}\right)^{\frac{2}{3}} - 81 x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + 27 \left(1 - x^{2}\right)^{\frac{2}{3}}} + \frac{40 x}{3 \left(- x^{2} \left(1 - x^{2}\right)^{\frac{2}{3}} + \left(1 - x^{2}\right)^{\frac{2}{3}}\right)} - 3 \sin^{2}{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} + 6 \sin{\left(x \right)} \cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \sin{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \sin{\left(\sin{\left(x \right)} \right)} \cos^{4}{\left(x \right)} + 4 \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{120 x - 24}\right)$$
=
$$0$$
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]
/ ________ \
| 3 / 2 |
|-x*\/ 1 - x + sin(sin(x))|
lim |----------------------------|
x->0+| 4 |
\ x *(-1 + x) /
$$\lim_{x \to 0^+}\left(\frac{- x \sqrt[3]{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4} \left(x - 1\right)}\right)$$
0
$$0$$
= -4.65613439965245e-31
/ ________ \
| 3 / 2 |
|-x*\/ 1 - x + sin(sin(x))|
lim |----------------------------|
x->0-| 4 |
\ x *(-1 + x) /
$$\lim_{x \to 0^-}\left(\frac{- x \sqrt[3]{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4} \left(x - 1\right)}\right)$$
0
$$0$$
= 2.42311962165042e-29
= 2.42311962165042e-29
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{- x \sqrt[3]{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4} \left(x - 1\right)}\right) = 0$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{- x \sqrt[3]{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4} \left(x - 1\right)}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{- x \sqrt[3]{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4} \left(x - 1\right)}\right) = 0$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{- x \sqrt[3]{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4} \left(x - 1\right)}\right) = -\infty$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{- x \sqrt[3]{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4} \left(x - 1\right)}\right) = \infty$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{- x \sqrt[3]{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4} \left(x - 1\right)}\right) = 0$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{- x \sqrt[3]{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4} \left(x - 1\right)}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{- x \sqrt[3]{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4} \left(x - 1\right)}\right) = 0$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{- x \sqrt[3]{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4} \left(x - 1\right)}\right) = -\infty$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{- x \sqrt[3]{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4} \left(x - 1\right)}\right) = \infty$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{- x \sqrt[3]{1 - x^{2}} + \sin{\left(\sin{\left(x \right)} \right)}}{x^{4} \left(x - 1\right)}\right) = 0$$
Más detalles con x→-oo