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Límite de la función (-sin(x)+atan(x))/x^5
Solución
Ha introducido
[src]
/-sin(x) + atan(x)\
lim |-----------------|
x->0+| 5 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} + \operatorname{atan}{\left(x \right)}}{x^{5}}\right)$$
Limit((-sin(x) + atan(x))/x^5, x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(- \sin{\left(x \right)} + \operatorname{atan}{\left(x \right)}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} x^{5} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} + \operatorname{atan}{\left(x \right)}}{x^{5}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \sin{\left(x \right)} + \operatorname{atan}{\left(x \right)}\right)}{\frac{d}{d x} x^{5}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \cos{\left(x \right)} + \frac{1}{x^{2} + 1}}{5 x^{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \cos{\left(x \right)} + \frac{1}{x^{2} + 1}\right)}{\frac{d}{d x} 5 x^{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{2 x}{x^{4} + 2 x^{2} + 1} + \sin{\left(x \right)}}{20 x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{2 x}{x^{4} + 2 x^{2} + 1} + \sin{\left(x \right)}\right)}{\frac{d}{d x} 20 x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{8 x^{4}}{x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1} + \frac{8 x^{2}}{x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1} + \cos{\left(x \right)} - \frac{2}{x^{4} + 2 x^{2} + 1}}{60 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\frac{8 x^{4}}{x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1} + \frac{8 x^{2}}{x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1} + \cos{\left(x \right)} - \frac{2}{x^{4} + 2 x^{2} + 1}\right)}{\frac{d}{d x} 60 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{64 x^{11}}{x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1} - \frac{256 x^{9}}{x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1} - \frac{384 x^{7}}{x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1} - \frac{256 x^{5}}{x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1} - \frac{64 x^{3}}{x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1} + \frac{40 x^{3}}{x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1} + \frac{24 x}{x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1} - \sin{\left(x \right)}}{120 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{64 x^{11}}{x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1} - \frac{256 x^{9}}{x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1} - \frac{384 x^{7}}{x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1} - \frac{256 x^{5}}{x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1} - \frac{64 x^{3}}{x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1} + \frac{40 x^{3}}{x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1} + \frac{24 x}{x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1} - \sin{\left(x \right)}\right)}{\frac{d}{d x} 120 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{128 x^{26}}{15 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} + \frac{1408 x^{24}}{15 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} + \frac{1408 x^{22}}{3 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} + \frac{1408 x^{20}}{x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1} + \frac{2816 x^{18}}{x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1} + \frac{19712 x^{16}}{5 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} + \frac{19712 x^{14}}{5 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} + \frac{2816 x^{12}}{x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1} + \frac{1408 x^{10}}{x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1} - \frac{128 x^{10}}{15 \left(x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1\right)} + \frac{1408 x^{8}}{3 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} - \frac{144 x^{8}}{5 \left(x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1\right)} + \frac{1408 x^{6}}{15 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} - \frac{176 x^{6}}{5 \left(x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1\right)} + \frac{128 x^{4}}{15 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} - \frac{272 x^{4}}{15 \left(x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1\right)} - \frac{16 x^{2}}{5 \left(x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1\right)} + \frac{x^{2}}{x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1} - \frac{\cos{\left(x \right)}}{120} + \frac{1}{5 \left(x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{128 x^{26}}{15 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} + \frac{1408 x^{24}}{15 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} + \frac{1408 x^{22}}{3 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} + \frac{1408 x^{20}}{x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1} + \frac{2816 x^{18}}{x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1} + \frac{19712 x^{16}}{5 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} + \frac{19712 x^{14}}{5 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} + \frac{2816 x^{12}}{x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1} + \frac{1408 x^{10}}{x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1} - \frac{128 x^{10}}{15 \left(x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1\right)} + \frac{1408 x^{8}}{3 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} - \frac{144 x^{8}}{5 \left(x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1\right)} + \frac{1408 x^{6}}{15 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} - \frac{176 x^{6}}{5 \left(x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1\right)} + \frac{128 x^{4}}{15 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} - \frac{272 x^{4}}{15 \left(x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1\right)} - \frac{16 x^{2}}{5 \left(x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1\right)} + \frac{x^{2}}{x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1} - \frac{\cos{\left(x \right)}}{120} + \frac{1}{5 \left(x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1\right)}\right)$$
=
$$-\infty$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 5 vez (veces)
0/0,
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(- \sin{\left(x \right)} + \operatorname{atan}{\left(x \right)}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} x^{5} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} + \operatorname{atan}{\left(x \right)}}{x^{5}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \sin{\left(x \right)} + \operatorname{atan}{\left(x \right)}\right)}{\frac{d}{d x} x^{5}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \cos{\left(x \right)} + \frac{1}{x^{2} + 1}}{5 x^{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \cos{\left(x \right)} + \frac{1}{x^{2} + 1}\right)}{\frac{d}{d x} 5 x^{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{2 x}{x^{4} + 2 x^{2} + 1} + \sin{\left(x \right)}}{20 x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{2 x}{x^{4} + 2 x^{2} + 1} + \sin{\left(x \right)}\right)}{\frac{d}{d x} 20 x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{8 x^{4}}{x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1} + \frac{8 x^{2}}{x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1} + \cos{\left(x \right)} - \frac{2}{x^{4} + 2 x^{2} + 1}}{60 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\frac{8 x^{4}}{x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1} + \frac{8 x^{2}}{x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1} + \cos{\left(x \right)} - \frac{2}{x^{4} + 2 x^{2} + 1}\right)}{\frac{d}{d x} 60 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{64 x^{11}}{x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1} - \frac{256 x^{9}}{x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1} - \frac{384 x^{7}}{x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1} - \frac{256 x^{5}}{x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1} - \frac{64 x^{3}}{x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1} + \frac{40 x^{3}}{x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1} + \frac{24 x}{x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1} - \sin{\left(x \right)}}{120 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{64 x^{11}}{x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1} - \frac{256 x^{9}}{x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1} - \frac{384 x^{7}}{x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1} - \frac{256 x^{5}}{x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1} - \frac{64 x^{3}}{x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1} + \frac{40 x^{3}}{x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1} + \frac{24 x}{x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1} - \sin{\left(x \right)}\right)}{\frac{d}{d x} 120 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{128 x^{26}}{15 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} + \frac{1408 x^{24}}{15 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} + \frac{1408 x^{22}}{3 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} + \frac{1408 x^{20}}{x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1} + \frac{2816 x^{18}}{x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1} + \frac{19712 x^{16}}{5 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} + \frac{19712 x^{14}}{5 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} + \frac{2816 x^{12}}{x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1} + \frac{1408 x^{10}}{x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1} - \frac{128 x^{10}}{15 \left(x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1\right)} + \frac{1408 x^{8}}{3 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} - \frac{144 x^{8}}{5 \left(x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1\right)} + \frac{1408 x^{6}}{15 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} - \frac{176 x^{6}}{5 \left(x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1\right)} + \frac{128 x^{4}}{15 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} - \frac{272 x^{4}}{15 \left(x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1\right)} - \frac{16 x^{2}}{5 \left(x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1\right)} + \frac{x^{2}}{x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1} - \frac{\cos{\left(x \right)}}{120} + \frac{1}{5 \left(x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{128 x^{26}}{15 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} + \frac{1408 x^{24}}{15 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} + \frac{1408 x^{22}}{3 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} + \frac{1408 x^{20}}{x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1} + \frac{2816 x^{18}}{x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1} + \frac{19712 x^{16}}{5 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} + \frac{19712 x^{14}}{5 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} + \frac{2816 x^{12}}{x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1} + \frac{1408 x^{10}}{x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1} - \frac{128 x^{10}}{15 \left(x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1\right)} + \frac{1408 x^{8}}{3 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} - \frac{144 x^{8}}{5 \left(x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1\right)} + \frac{1408 x^{6}}{15 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} - \frac{176 x^{6}}{5 \left(x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1\right)} + \frac{128 x^{4}}{15 \left(x^{32} + 16 x^{30} + 120 x^{28} + 560 x^{26} + 1820 x^{24} + 4368 x^{22} + 8008 x^{20} + 11440 x^{18} + 12870 x^{16} + 11440 x^{14} + 8008 x^{12} + 4368 x^{10} + 1820 x^{8} + 560 x^{6} + 120 x^{4} + 16 x^{2} + 1\right)} - \frac{272 x^{4}}{15 \left(x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1\right)} - \frac{16 x^{2}}{5 \left(x^{16} + 8 x^{14} + 28 x^{12} + 56 x^{10} + 70 x^{8} + 56 x^{6} + 28 x^{4} + 8 x^{2} + 1\right)} + \frac{x^{2}}{x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1} - \frac{\cos{\left(x \right)}}{120} + \frac{1}{5 \left(x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1\right)}\right)$$
=
$$-\infty$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 5 vez (veces)
Gráfica
A la izquierda y a la derecha
[src]
/-sin(x) + atan(x)\
lim |-----------------|
x->0+| 5 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} + \operatorname{atan}{\left(x \right)}}{x^{5}}\right)$$
-oo
$$-\infty$$
= -3799.97500625647
/-sin(x) + atan(x)\
lim |-----------------|
x->0-| 5 |
\ x /
$$\lim_{x \to 0^-}\left(\frac{- \sin{\left(x \right)} + \operatorname{atan}{\left(x \right)}}{x^{5}}\right)$$
-oo
$$-\infty$$
= -3799.97500625647
= -3799.97500625647
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{- \sin{\left(x \right)} + \operatorname{atan}{\left(x \right)}}{x^{5}}\right) = -\infty$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} + \operatorname{atan}{\left(x \right)}}{x^{5}}\right) = -\infty$$
$$\lim_{x \to \infty}\left(\frac{- \sin{\left(x \right)} + \operatorname{atan}{\left(x \right)}}{x^{5}}\right) = 0$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{- \sin{\left(x \right)} + \operatorname{atan}{\left(x \right)}}{x^{5}}\right) = - \sin{\left(1 \right)} + \frac{\pi}{4}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{- \sin{\left(x \right)} + \operatorname{atan}{\left(x \right)}}{x^{5}}\right) = - \sin{\left(1 \right)} + \frac{\pi}{4}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{- \sin{\left(x \right)} + \operatorname{atan}{\left(x \right)}}{x^{5}}\right) = 0$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} + \operatorname{atan}{\left(x \right)}}{x^{5}}\right) = -\infty$$
$$\lim_{x \to \infty}\left(\frac{- \sin{\left(x \right)} + \operatorname{atan}{\left(x \right)}}{x^{5}}\right) = 0$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{- \sin{\left(x \right)} + \operatorname{atan}{\left(x \right)}}{x^{5}}\right) = - \sin{\left(1 \right)} + \frac{\pi}{4}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{- \sin{\left(x \right)} + \operatorname{atan}{\left(x \right)}}{x^{5}}\right) = - \sin{\left(1 \right)} + \frac{\pi}{4}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{- \sin{\left(x \right)} + \operatorname{atan}{\left(x \right)}}{x^{5}}\right) = 0$$
Más detalles con x→-oo