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Expresiones semejantes
- -1+e^(x^3)+x^3/tan(x/2)^6
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Expresiones con funciones
- Tangente tan
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Límite de la función -1+e^(x^3)-x^3/tan(x/2)^6
Solución
Ha introducido
[src]
/ / 3\ 3 \
| \x / x |
lim |-1 + E - -------|
x->0+| 6/x\|
| tan |-||
\ \2//
$$\lim_{x \to 0^+}\left(- \frac{x^{3}}{\tan^{6}{\left(\frac{x}{2} \right)}} + \left(e^{x^{3}} - 1\right)\right)$$
Limit(-1 + E^(x^3) - x^3/tan(x/2)^6, x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(- x^{3} + e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} - \tan^{6}{\left(\frac{x}{2} \right)}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \tan^{6}{\left(\frac{x}{2} \right)} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(- \frac{x^{3}}{\tan^{6}{\left(\frac{x}{2} \right)}} + \left(e^{x^{3}} - 1\right)\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\frac{- x^{3} + \left(e^{x^{3}} - 1\right) \tan^{6}{\left(\frac{x}{2} \right)}}{\tan^{6}{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- x^{3} + e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} - \tan^{6}{\left(\frac{x}{2} \right)}\right)}{\frac{d}{d x} \tan^{6}{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{3 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} - 3 x^{2} + \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} - \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) \tan^{5}{\left(\frac{x}{2} \right)}}{\left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) \tan^{5}{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{3 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} - 3 x^{2} + 3 e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 3 e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} - 3 \tan^{7}{\left(\frac{x}{2} \right)} - 3 \tan^{5}{\left(\frac{x}{2} \right)}}{3 \tan^{5}{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(3 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} - 3 x^{2} + 3 e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 3 e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} - 3 \tan^{7}{\left(\frac{x}{2} \right)} - 3 \tan^{5}{\left(\frac{x}{2} \right)}\right)}{\frac{d}{d x} 3 \tan^{5}{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{9 x^{4} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 3 x^{2} \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 9 x^{2} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 9 x^{2} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 6 x e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} - 6 x + 3 \left(\frac{5 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{5}{2}\right) e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} - 3 \left(\frac{5 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{5}{2}\right) \tan^{4}{\left(\frac{x}{2} \right)} + 3 \left(\frac{7 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{7}{2}\right) e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} - 3 \left(\frac{7 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{7}{2}\right) \tan^{6}{\left(\frac{x}{2} \right)}}{3 \left(\frac{5 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{5}{2}\right) \tan^{4}{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \left(9 x^{4} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 18 x^{2} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 18 x^{2} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 6 x e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} - 6 x + \frac{21 e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)}}{2} + 18 e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{15 e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)}}{2} - \frac{21 \tan^{8}{\left(\frac{x}{2} \right)}}{2} - 18 \tan^{6}{\left(\frac{x}{2} \right)} - \frac{15 \tan^{4}{\left(\frac{x}{2} \right)}}{2}\right)}{15 \tan^{4}{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(9 x^{4} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 18 x^{2} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 18 x^{2} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 6 x e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} - 6 x + \frac{21 e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)}}{2} + 18 e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{15 e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)}}{2} - \frac{21 \tan^{8}{\left(\frac{x}{2} \right)}}{2} - 18 \tan^{6}{\left(\frac{x}{2} \right)} - \frac{15 \tan^{4}{\left(\frac{x}{2} \right)}}{2}\right)}{\frac{d}{d x} \frac{15 \tan^{4}{\left(\frac{x}{2} \right)}}{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \left(27 x^{6} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 9 x^{4} \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 54 x^{4} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 54 x^{4} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 54 x^{3} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 18 x^{2} \left(\frac{5 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{5}{2}\right) e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 18 x^{2} \left(\frac{7 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{7}{2}\right) e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{63 x^{2} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)}}{2} + 54 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{45 x^{2} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)}}{2} + 6 x \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 36 x e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 36 x e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + \frac{15 \left(2 \tan^{2}{\left(\frac{x}{2} \right)} + 2\right) e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)}}{2} - \frac{15 \left(2 \tan^{2}{\left(\frac{x}{2} \right)} + 2\right) \tan^{3}{\left(\frac{x}{2} \right)}}{2} + 18 \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} - 18 \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) \tan^{5}{\left(\frac{x}{2} \right)} + \frac{21 \left(4 \tan^{2}{\left(\frac{x}{2} \right)} + 4\right) e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)}}{2} - \frac{21 \left(4 \tan^{2}{\left(\frac{x}{2} \right)} + 4\right) \tan^{7}{\left(\frac{x}{2} \right)}}{2} + 6 e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} - 6\right)}{15 \left(2 \tan^{2}{\left(\frac{x}{2} \right)} + 2\right) \tan^{3}{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{27 x^{6} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 81 x^{4} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 81 x^{4} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 54 x^{3} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{189 x^{2} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)}}{2} + 162 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{135 x^{2} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)}}{2} + 54 x e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 54 x e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 42 e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 96 e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 6 e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 69 e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 15 e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} - 42 \tan^{9}{\left(\frac{x}{2} \right)} - 96 \tan^{7}{\left(\frac{x}{2} \right)} - 69 \tan^{5}{\left(\frac{x}{2} \right)} - 15 \tan^{3}{\left(\frac{x}{2} \right)} - 6}{15 \tan^{3}{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(27 x^{6} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 81 x^{4} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 81 x^{4} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 54 x^{3} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{189 x^{2} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)}}{2} + 162 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{135 x^{2} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)}}{2} + 54 x e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 54 x e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 42 e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 96 e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 6 e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 69 e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 15 e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} - 42 \tan^{9}{\left(\frac{x}{2} \right)} - 96 \tan^{7}{\left(\frac{x}{2} \right)} - 69 \tan^{5}{\left(\frac{x}{2} \right)} - 15 \tan^{3}{\left(\frac{x}{2} \right)} - 6\right)}{\frac{d}{d x} 15 \tan^{3}{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{81 x^{8} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 27 x^{6} \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 243 x^{6} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 243 x^{6} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 324 x^{5} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 81 x^{4} \left(\frac{5 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{5}{2}\right) e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 81 x^{4} \left(\frac{7 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{7}{2}\right) e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{567 x^{4} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)}}{2} + 486 x^{4} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{405 x^{4} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)}}{2} + 54 x^{3} \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 486 x^{3} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 486 x^{3} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + \frac{135 x^{2} \left(2 \tan^{2}{\left(\frac{x}{2} \right)} + 2\right) e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)}}{2} + 162 x^{2} \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + \frac{189 x^{2} \left(4 \tan^{2}{\left(\frac{x}{2} \right)} + 4\right) e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)}}{2} + 126 x^{2} e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 288 x^{2} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 180 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 207 x^{2} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 45 x^{2} e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 54 x \left(\frac{5 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{5}{2}\right) e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 54 x \left(\frac{7 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{7}{2}\right) e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 189 x e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 324 x e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 135 x e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 15 \left(\frac{3 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{3}{2}\right) e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)} - 15 \left(\frac{3 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{3}{2}\right) \tan^{2}{\left(\frac{x}{2} \right)} + 69 \left(\frac{5 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{5}{2}\right) e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} - 69 \left(\frac{5 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{5}{2}\right) \tan^{4}{\left(\frac{x}{2} \right)} + 6 \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 96 \left(\frac{7 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{7}{2}\right) e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} - 96 \left(\frac{7 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{7}{2}\right) \tan^{6}{\left(\frac{x}{2} \right)} + 42 \left(\frac{9 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{9}{2}\right) e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} - 42 \left(\frac{9 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{9}{2}\right) \tan^{8}{\left(\frac{x}{2} \right)} + 54 e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 54 e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)}}{15 \left(\frac{3 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{3}{2}\right) \tan^{2}{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \left(81 x^{8} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 324 x^{6} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 324 x^{6} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 324 x^{5} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 567 x^{4} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 972 x^{4} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 405 x^{4} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 648 x^{3} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 648 x^{3} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 504 x^{2} e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 1152 x^{2} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 180 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 828 x^{2} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 180 x^{2} e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 378 x e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 648 x e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 270 x e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 189 e^{x^{3}} \tan^{10}{\left(\frac{x}{2} \right)} + 525 e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 72 e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + \frac{1017 e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)}}{2} + 72 e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 195 e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + \frac{45 e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)}}{2} - 189 \tan^{10}{\left(\frac{x}{2} \right)} - 525 \tan^{8}{\left(\frac{x}{2} \right)} - \frac{1017 \tan^{6}{\left(\frac{x}{2} \right)}}{2} - 195 \tan^{4}{\left(\frac{x}{2} \right)} - \frac{45 \tan^{2}{\left(\frac{x}{2} \right)}}{2}\right)}{45 \tan^{2}{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(81 x^{8} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 324 x^{6} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 324 x^{6} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 324 x^{5} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 567 x^{4} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 972 x^{4} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 405 x^{4} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 648 x^{3} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 648 x^{3} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 504 x^{2} e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 1152 x^{2} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 180 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 828 x^{2} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 180 x^{2} e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 378 x e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 648 x e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 270 x e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 189 e^{x^{3}} \tan^{10}{\left(\frac{x}{2} \right)} + 525 e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 72 e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + \frac{1017 e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)}}{2} + 72 e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 195 e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + \frac{45 e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)}}{2} - 189 \tan^{10}{\left(\frac{x}{2} \right)} - 525 \tan^{8}{\left(\frac{x}{2} \right)} - \frac{1017 \tan^{6}{\left(\frac{x}{2} \right)}}{2} - 195 \tan^{4}{\left(\frac{x}{2} \right)} - \frac{45 \tan^{2}{\left(\frac{x}{2} \right)}}{2}\right)}{\frac{d}{d x} \frac{45 \tan^{2}{\left(\frac{x}{2} \right)}}{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \left(243 x^{10} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 81 x^{8} \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 972 x^{8} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 972 x^{8} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 1620 x^{7} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 324 x^{6} \left(\frac{5 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{5}{2}\right) e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 324 x^{6} \left(\frac{7 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{7}{2}\right) e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 1701 x^{6} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 2916 x^{6} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 1215 x^{6} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 324 x^{5} \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 3888 x^{5} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 3888 x^{5} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 405 x^{4} \left(2 \tan^{2}{\left(\frac{x}{2} \right)} + 2\right) e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 972 x^{4} \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 567 x^{4} \left(4 \tan^{2}{\left(\frac{x}{2} \right)} + 4\right) e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 1512 x^{4} e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 3456 x^{4} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 2160 x^{4} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 2484 x^{4} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 540 x^{4} e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 648 x^{3} \left(\frac{5 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{5}{2}\right) e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 648 x^{3} \left(\frac{7 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{7}{2}\right) e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 3402 x^{3} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 5832 x^{3} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 2430 x^{3} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 180 x^{2} \left(\frac{3 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{3}{2}\right) e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)} + 828 x^{2} \left(\frac{5 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{5}{2}\right) e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 180 x^{2} \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 1152 x^{2} \left(\frac{7 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{7}{2}\right) e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 504 x^{2} \left(\frac{9 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{9}{2}\right) e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 567 x^{2} e^{x^{3}} \tan^{10}{\left(\frac{x}{2} \right)} + 1575 x^{2} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 2160 x^{2} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + \frac{3051 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)}}{2} + 2160 x^{2} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 585 x^{2} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + \frac{135 x^{2} e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)}}{2} + 270 x \left(2 \tan^{2}{\left(\frac{x}{2} \right)} + 2\right) e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 648 x \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 378 x \left(4 \tan^{2}{\left(\frac{x}{2} \right)} + 4\right) e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 1008 x e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 2304 x e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 360 x e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 1656 x e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 360 x e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + \frac{45 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) e^{x^{3}} \tan{\left(\frac{x}{2} \right)}}{2} - \frac{45 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}{2} + 195 \left(2 \tan^{2}{\left(\frac{x}{2} \right)} + 2\right) e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} - 195 \left(2 \tan^{2}{\left(\frac{x}{2} \right)} + 2\right) \tan^{3}{\left(\frac{x}{2} \right)} + 72 \left(\frac{5 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{5}{2}\right) e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + \frac{1017 \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)}}{2} - \frac{1017 \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) \tan^{5}{\left(\frac{x}{2} \right)}}{2} + 72 \left(\frac{7 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{7}{2}\right) e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 525 \left(4 \tan^{2}{\left(\frac{x}{2} \right)} + 4\right) e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} - 525 \left(4 \tan^{2}{\left(\frac{x}{2} \right)} + 4\right) \tan^{7}{\left(\frac{x}{2} \right)} + 189 \left(5 \tan^{2}{\left(\frac{x}{2} \right)} + 5\right) e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} - 189 \left(5 \tan^{2}{\left(\frac{x}{2} \right)} + 5\right) \tan^{9}{\left(\frac{x}{2} \right)} + 378 e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 648 e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 270 e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)}\right)}{45 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \left(243 x^{10} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 1215 x^{8} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 1215 x^{8} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 1620 x^{7} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 2835 x^{6} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 4860 x^{6} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 2025 x^{6} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 4860 x^{5} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 4860 x^{5} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 3780 x^{4} e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 8640 x^{4} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 2160 x^{4} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 6210 x^{4} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 1350 x^{4} e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 5670 x^{3} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 9720 x^{3} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 4050 x^{3} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 2835 x^{2} e^{x^{3}} \tan^{10}{\left(\frac{x}{2} \right)} + 7875 x^{2} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 2700 x^{2} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + \frac{15255 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)}}{2} + 2700 x^{2} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 2925 x^{2} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + \frac{675 x^{2} e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)}}{2} + 2520 x e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 5760 x e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 360 x e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 4140 x e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 900 x e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 945 e^{x^{3}} \tan^{11}{\left(\frac{x}{2} \right)} + 3045 e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 630 e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + \frac{7251 e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)}}{2} + 1080 e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{3831 e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)}}{2} + 450 e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + \frac{825 e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)}}{2} + \frac{45 e^{x^{3}} \tan{\left(\frac{x}{2} \right)}}{2} - 945 \tan^{11}{\left(\frac{x}{2} \right)} - 3045 \tan^{9}{\left(\frac{x}{2} \right)} - \frac{7251 \tan^{7}{\left(\frac{x}{2} \right)}}{2} - \frac{3831 \tan^{5}{\left(\frac{x}{2} \right)}}{2} - \frac{825 \tan^{3}{\left(\frac{x}{2} \right)}}{2} - \frac{45 \tan{\left(\frac{x}{2} \right)}}{2}\right)}{45 \tan{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(243 x^{10} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 1215 x^{8} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 1215 x^{8} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 1620 x^{7} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 2835 x^{6} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 4860 x^{6} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 2025 x^{6} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 4860 x^{5} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 4860 x^{5} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 3780 x^{4} e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 8640 x^{4} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 2160 x^{4} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 6210 x^{4} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 1350 x^{4} e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 5670 x^{3} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 9720 x^{3} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 4050 x^{3} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 2835 x^{2} e^{x^{3}} \tan^{10}{\left(\frac{x}{2} \right)} + 7875 x^{2} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 2700 x^{2} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + \frac{15255 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)}}{2} + 2700 x^{2} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 2925 x^{2} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + \frac{675 x^{2} e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)}}{2} + 2520 x e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 5760 x e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 360 x e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 4140 x e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 900 x e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 945 e^{x^{3}} \tan^{11}{\left(\frac{x}{2} \right)} + 3045 e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 630 e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + \frac{7251 e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)}}{2} + 1080 e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{3831 e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)}}{2} + 450 e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + \frac{825 e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)}}{2} + \frac{45 e^{x^{3}} \tan{\left(\frac{x}{2} \right)}}{2} - 945 \tan^{11}{\left(\frac{x}{2} \right)} - 3045 \tan^{9}{\left(\frac{x}{2} \right)} - \frac{7251 \tan^{7}{\left(\frac{x}{2} \right)}}{2} - \frac{3831 \tan^{5}{\left(\frac{x}{2} \right)}}{2} - \frac{825 \tan^{3}{\left(\frac{x}{2} \right)}}{2} - \frac{45 \tan{\left(\frac{x}{2} \right)}}{2}\right)}{\frac{d}{d x} \frac{45 \tan{\left(\frac{x}{2} \right)}}{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{729 x^{12} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 4374 x^{10} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 4374 x^{10} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 7290 x^{9} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{25515 x^{8} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)}}{2} + 21870 x^{8} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{18225 x^{8} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)}}{2} + 29160 x^{7} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 29160 x^{7} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 22680 x^{6} e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 51840 x^{6} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 17820 x^{6} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 37260 x^{6} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 8100 x^{6} e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 51030 x^{5} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 87480 x^{5} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 36450 x^{5} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 25515 x^{4} e^{x^{3}} \tan^{10}{\left(\frac{x}{2} \right)} + 70875 x^{4} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 38880 x^{4} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + \frac{137295 x^{4} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)}}{2} + 38880 x^{4} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 26325 x^{4} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + \frac{6075 x^{4} e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)}}{2} + 45360 x^{3} e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 103680 x^{3} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 9720 x^{3} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 74520 x^{3} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 16200 x^{3} e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 17010 x^{2} e^{x^{3}} \tan^{11}{\left(\frac{x}{2} \right)} + 54810 x^{2} e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 28350 x^{2} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 65259 x^{2} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 48600 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 34479 x^{2} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 20250 x^{2} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 7425 x^{2} e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 405 x^{2} e^{x^{3}} \tan{\left(\frac{x}{2} \right)} + 17010 x e^{x^{3}} \tan^{10}{\left(\frac{x}{2} \right)} + 47250 x e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 6480 x e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 45765 x e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 6480 x e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 17550 x e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 2025 x e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)} + \frac{10395 e^{x^{3}} \tan^{12}{\left(\frac{x}{2} \right)}}{2} + 18900 e^{x^{3}} \tan^{10}{\left(\frac{x}{2} \right)} + 5040 e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + \frac{105567 e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)}}{4} + 11520 e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 17838 e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 8280 e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + \frac{10815 e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)}}{2} + 1800 e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 630 e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)} + \frac{45 e^{x^{3}}}{4} - \frac{10395 \tan^{12}{\left(\frac{x}{2} \right)}}{2} - 18900 \tan^{10}{\left(\frac{x}{2} \right)} - \frac{105567 \tan^{8}{\left(\frac{x}{2} \right)}}{4} - 17478 \tan^{6}{\left(\frac{x}{2} \right)} - \frac{10815 \tan^{4}{\left(\frac{x}{2} \right)}}{2} - 630 \tan^{2}{\left(\frac{x}{2} \right)} - \frac{45}{4}}{\frac{45 \tan^{2}{\left(\frac{x}{2} \right)}}{4} + \frac{45}{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{729 x^{12} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 4374 x^{10} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 4374 x^{10} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 7290 x^{9} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{25515 x^{8} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)}}{2} + 21870 x^{8} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{18225 x^{8} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)}}{2} + 29160 x^{7} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 29160 x^{7} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 22680 x^{6} e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 51840 x^{6} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 17820 x^{6} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 37260 x^{6} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 8100 x^{6} e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 51030 x^{5} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 87480 x^{5} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 36450 x^{5} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 25515 x^{4} e^{x^{3}} \tan^{10}{\left(\frac{x}{2} \right)} + 70875 x^{4} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 38880 x^{4} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + \frac{137295 x^{4} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)}}{2} + 38880 x^{4} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 26325 x^{4} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + \frac{6075 x^{4} e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)}}{2} + 45360 x^{3} e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 103680 x^{3} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 9720 x^{3} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 74520 x^{3} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 16200 x^{3} e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 17010 x^{2} e^{x^{3}} \tan^{11}{\left(\frac{x}{2} \right)} + 54810 x^{2} e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 28350 x^{2} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 65259 x^{2} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 48600 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 34479 x^{2} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 20250 x^{2} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 7425 x^{2} e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 405 x^{2} e^{x^{3}} \tan{\left(\frac{x}{2} \right)} + 17010 x e^{x^{3}} \tan^{10}{\left(\frac{x}{2} \right)} + 47250 x e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 6480 x e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 45765 x e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 6480 x e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 17550 x e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 2025 x e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)} + \frac{10395 e^{x^{3}} \tan^{12}{\left(\frac{x}{2} \right)}}{2} + 18900 e^{x^{3}} \tan^{10}{\left(\frac{x}{2} \right)} + 5040 e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + \frac{105567 e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)}}{4} + 11520 e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 17838 e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 8280 e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + \frac{10815 e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)}}{2} + 1800 e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 630 e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)} + \frac{45 e^{x^{3}}}{4} - \frac{10395 \tan^{12}{\left(\frac{x}{2} \right)}}{2} - 18900 \tan^{10}{\left(\frac{x}{2} \right)} - \frac{105567 \tan^{8}{\left(\frac{x}{2} \right)}}{4} - 17478 \tan^{6}{\left(\frac{x}{2} \right)} - \frac{10815 \tan^{4}{\left(\frac{x}{2} \right)}}{2} - 630 \tan^{2}{\left(\frac{x}{2} \right)} - \frac{45}{4}}{\frac{45 \tan^{2}{\left(\frac{x}{2} \right)}}{4} + \frac{45}{4}}\right)$$
=
$$-\infty$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 6 vez (veces)
0/0,
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(- x^{3} + e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} - \tan^{6}{\left(\frac{x}{2} \right)}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \tan^{6}{\left(\frac{x}{2} \right)} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(- \frac{x^{3}}{\tan^{6}{\left(\frac{x}{2} \right)}} + \left(e^{x^{3}} - 1\right)\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\frac{- x^{3} + \left(e^{x^{3}} - 1\right) \tan^{6}{\left(\frac{x}{2} \right)}}{\tan^{6}{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- x^{3} + e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} - \tan^{6}{\left(\frac{x}{2} \right)}\right)}{\frac{d}{d x} \tan^{6}{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{3 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} - 3 x^{2} + \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} - \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) \tan^{5}{\left(\frac{x}{2} \right)}}{\left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) \tan^{5}{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{3 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} - 3 x^{2} + 3 e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 3 e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} - 3 \tan^{7}{\left(\frac{x}{2} \right)} - 3 \tan^{5}{\left(\frac{x}{2} \right)}}{3 \tan^{5}{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(3 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} - 3 x^{2} + 3 e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 3 e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} - 3 \tan^{7}{\left(\frac{x}{2} \right)} - 3 \tan^{5}{\left(\frac{x}{2} \right)}\right)}{\frac{d}{d x} 3 \tan^{5}{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{9 x^{4} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 3 x^{2} \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 9 x^{2} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 9 x^{2} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 6 x e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} - 6 x + 3 \left(\frac{5 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{5}{2}\right) e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} - 3 \left(\frac{5 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{5}{2}\right) \tan^{4}{\left(\frac{x}{2} \right)} + 3 \left(\frac{7 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{7}{2}\right) e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} - 3 \left(\frac{7 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{7}{2}\right) \tan^{6}{\left(\frac{x}{2} \right)}}{3 \left(\frac{5 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{5}{2}\right) \tan^{4}{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \left(9 x^{4} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 18 x^{2} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 18 x^{2} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 6 x e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} - 6 x + \frac{21 e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)}}{2} + 18 e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{15 e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)}}{2} - \frac{21 \tan^{8}{\left(\frac{x}{2} \right)}}{2} - 18 \tan^{6}{\left(\frac{x}{2} \right)} - \frac{15 \tan^{4}{\left(\frac{x}{2} \right)}}{2}\right)}{15 \tan^{4}{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(9 x^{4} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 18 x^{2} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 18 x^{2} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 6 x e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} - 6 x + \frac{21 e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)}}{2} + 18 e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{15 e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)}}{2} - \frac{21 \tan^{8}{\left(\frac{x}{2} \right)}}{2} - 18 \tan^{6}{\left(\frac{x}{2} \right)} - \frac{15 \tan^{4}{\left(\frac{x}{2} \right)}}{2}\right)}{\frac{d}{d x} \frac{15 \tan^{4}{\left(\frac{x}{2} \right)}}{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \left(27 x^{6} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 9 x^{4} \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 54 x^{4} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 54 x^{4} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 54 x^{3} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 18 x^{2} \left(\frac{5 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{5}{2}\right) e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 18 x^{2} \left(\frac{7 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{7}{2}\right) e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{63 x^{2} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)}}{2} + 54 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{45 x^{2} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)}}{2} + 6 x \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 36 x e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 36 x e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + \frac{15 \left(2 \tan^{2}{\left(\frac{x}{2} \right)} + 2\right) e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)}}{2} - \frac{15 \left(2 \tan^{2}{\left(\frac{x}{2} \right)} + 2\right) \tan^{3}{\left(\frac{x}{2} \right)}}{2} + 18 \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} - 18 \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) \tan^{5}{\left(\frac{x}{2} \right)} + \frac{21 \left(4 \tan^{2}{\left(\frac{x}{2} \right)} + 4\right) e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)}}{2} - \frac{21 \left(4 \tan^{2}{\left(\frac{x}{2} \right)} + 4\right) \tan^{7}{\left(\frac{x}{2} \right)}}{2} + 6 e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} - 6\right)}{15 \left(2 \tan^{2}{\left(\frac{x}{2} \right)} + 2\right) \tan^{3}{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{27 x^{6} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 81 x^{4} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 81 x^{4} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 54 x^{3} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{189 x^{2} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)}}{2} + 162 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{135 x^{2} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)}}{2} + 54 x e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 54 x e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 42 e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 96 e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 6 e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 69 e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 15 e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} - 42 \tan^{9}{\left(\frac{x}{2} \right)} - 96 \tan^{7}{\left(\frac{x}{2} \right)} - 69 \tan^{5}{\left(\frac{x}{2} \right)} - 15 \tan^{3}{\left(\frac{x}{2} \right)} - 6}{15 \tan^{3}{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(27 x^{6} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 81 x^{4} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 81 x^{4} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 54 x^{3} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{189 x^{2} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)}}{2} + 162 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{135 x^{2} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)}}{2} + 54 x e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 54 x e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 42 e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 96 e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 6 e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 69 e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 15 e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} - 42 \tan^{9}{\left(\frac{x}{2} \right)} - 96 \tan^{7}{\left(\frac{x}{2} \right)} - 69 \tan^{5}{\left(\frac{x}{2} \right)} - 15 \tan^{3}{\left(\frac{x}{2} \right)} - 6\right)}{\frac{d}{d x} 15 \tan^{3}{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{81 x^{8} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 27 x^{6} \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 243 x^{6} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 243 x^{6} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 324 x^{5} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 81 x^{4} \left(\frac{5 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{5}{2}\right) e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 81 x^{4} \left(\frac{7 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{7}{2}\right) e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{567 x^{4} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)}}{2} + 486 x^{4} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{405 x^{4} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)}}{2} + 54 x^{3} \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 486 x^{3} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 486 x^{3} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + \frac{135 x^{2} \left(2 \tan^{2}{\left(\frac{x}{2} \right)} + 2\right) e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)}}{2} + 162 x^{2} \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + \frac{189 x^{2} \left(4 \tan^{2}{\left(\frac{x}{2} \right)} + 4\right) e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)}}{2} + 126 x^{2} e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 288 x^{2} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 180 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 207 x^{2} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 45 x^{2} e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 54 x \left(\frac{5 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{5}{2}\right) e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 54 x \left(\frac{7 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{7}{2}\right) e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 189 x e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 324 x e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 135 x e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 15 \left(\frac{3 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{3}{2}\right) e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)} - 15 \left(\frac{3 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{3}{2}\right) \tan^{2}{\left(\frac{x}{2} \right)} + 69 \left(\frac{5 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{5}{2}\right) e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} - 69 \left(\frac{5 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{5}{2}\right) \tan^{4}{\left(\frac{x}{2} \right)} + 6 \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 96 \left(\frac{7 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{7}{2}\right) e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} - 96 \left(\frac{7 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{7}{2}\right) \tan^{6}{\left(\frac{x}{2} \right)} + 42 \left(\frac{9 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{9}{2}\right) e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} - 42 \left(\frac{9 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{9}{2}\right) \tan^{8}{\left(\frac{x}{2} \right)} + 54 e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 54 e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)}}{15 \left(\frac{3 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{3}{2}\right) \tan^{2}{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \left(81 x^{8} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 324 x^{6} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 324 x^{6} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 324 x^{5} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 567 x^{4} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 972 x^{4} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 405 x^{4} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 648 x^{3} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 648 x^{3} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 504 x^{2} e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 1152 x^{2} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 180 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 828 x^{2} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 180 x^{2} e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 378 x e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 648 x e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 270 x e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 189 e^{x^{3}} \tan^{10}{\left(\frac{x}{2} \right)} + 525 e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 72 e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + \frac{1017 e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)}}{2} + 72 e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 195 e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + \frac{45 e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)}}{2} - 189 \tan^{10}{\left(\frac{x}{2} \right)} - 525 \tan^{8}{\left(\frac{x}{2} \right)} - \frac{1017 \tan^{6}{\left(\frac{x}{2} \right)}}{2} - 195 \tan^{4}{\left(\frac{x}{2} \right)} - \frac{45 \tan^{2}{\left(\frac{x}{2} \right)}}{2}\right)}{45 \tan^{2}{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(81 x^{8} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 324 x^{6} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 324 x^{6} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 324 x^{5} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 567 x^{4} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 972 x^{4} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 405 x^{4} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 648 x^{3} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 648 x^{3} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 504 x^{2} e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 1152 x^{2} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 180 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 828 x^{2} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 180 x^{2} e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 378 x e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 648 x e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 270 x e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 189 e^{x^{3}} \tan^{10}{\left(\frac{x}{2} \right)} + 525 e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 72 e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + \frac{1017 e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)}}{2} + 72 e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 195 e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + \frac{45 e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)}}{2} - 189 \tan^{10}{\left(\frac{x}{2} \right)} - 525 \tan^{8}{\left(\frac{x}{2} \right)} - \frac{1017 \tan^{6}{\left(\frac{x}{2} \right)}}{2} - 195 \tan^{4}{\left(\frac{x}{2} \right)} - \frac{45 \tan^{2}{\left(\frac{x}{2} \right)}}{2}\right)}{\frac{d}{d x} \frac{45 \tan^{2}{\left(\frac{x}{2} \right)}}{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \left(243 x^{10} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 81 x^{8} \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 972 x^{8} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 972 x^{8} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 1620 x^{7} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 324 x^{6} \left(\frac{5 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{5}{2}\right) e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 324 x^{6} \left(\frac{7 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{7}{2}\right) e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 1701 x^{6} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 2916 x^{6} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 1215 x^{6} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 324 x^{5} \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 3888 x^{5} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 3888 x^{5} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 405 x^{4} \left(2 \tan^{2}{\left(\frac{x}{2} \right)} + 2\right) e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 972 x^{4} \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 567 x^{4} \left(4 \tan^{2}{\left(\frac{x}{2} \right)} + 4\right) e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 1512 x^{4} e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 3456 x^{4} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 2160 x^{4} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 2484 x^{4} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 540 x^{4} e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 648 x^{3} \left(\frac{5 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{5}{2}\right) e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 648 x^{3} \left(\frac{7 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{7}{2}\right) e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 3402 x^{3} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 5832 x^{3} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 2430 x^{3} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 180 x^{2} \left(\frac{3 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{3}{2}\right) e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)} + 828 x^{2} \left(\frac{5 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{5}{2}\right) e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 180 x^{2} \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 1152 x^{2} \left(\frac{7 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{7}{2}\right) e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 504 x^{2} \left(\frac{9 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{9}{2}\right) e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 567 x^{2} e^{x^{3}} \tan^{10}{\left(\frac{x}{2} \right)} + 1575 x^{2} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 2160 x^{2} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + \frac{3051 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)}}{2} + 2160 x^{2} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 585 x^{2} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + \frac{135 x^{2} e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)}}{2} + 270 x \left(2 \tan^{2}{\left(\frac{x}{2} \right)} + 2\right) e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 648 x \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 378 x \left(4 \tan^{2}{\left(\frac{x}{2} \right)} + 4\right) e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 1008 x e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 2304 x e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 360 x e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 1656 x e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 360 x e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + \frac{45 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) e^{x^{3}} \tan{\left(\frac{x}{2} \right)}}{2} - \frac{45 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}{2} + 195 \left(2 \tan^{2}{\left(\frac{x}{2} \right)} + 2\right) e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} - 195 \left(2 \tan^{2}{\left(\frac{x}{2} \right)} + 2\right) \tan^{3}{\left(\frac{x}{2} \right)} + 72 \left(\frac{5 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{5}{2}\right) e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + \frac{1017 \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)}}{2} - \frac{1017 \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 3\right) \tan^{5}{\left(\frac{x}{2} \right)}}{2} + 72 \left(\frac{7 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{7}{2}\right) e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 525 \left(4 \tan^{2}{\left(\frac{x}{2} \right)} + 4\right) e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} - 525 \left(4 \tan^{2}{\left(\frac{x}{2} \right)} + 4\right) \tan^{7}{\left(\frac{x}{2} \right)} + 189 \left(5 \tan^{2}{\left(\frac{x}{2} \right)} + 5\right) e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} - 189 \left(5 \tan^{2}{\left(\frac{x}{2} \right)} + 5\right) \tan^{9}{\left(\frac{x}{2} \right)} + 378 e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 648 e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 270 e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)}\right)}{45 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \left(243 x^{10} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 1215 x^{8} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 1215 x^{8} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 1620 x^{7} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 2835 x^{6} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 4860 x^{6} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 2025 x^{6} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 4860 x^{5} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 4860 x^{5} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 3780 x^{4} e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 8640 x^{4} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 2160 x^{4} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 6210 x^{4} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 1350 x^{4} e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 5670 x^{3} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 9720 x^{3} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 4050 x^{3} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 2835 x^{2} e^{x^{3}} \tan^{10}{\left(\frac{x}{2} \right)} + 7875 x^{2} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 2700 x^{2} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + \frac{15255 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)}}{2} + 2700 x^{2} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 2925 x^{2} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + \frac{675 x^{2} e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)}}{2} + 2520 x e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 5760 x e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 360 x e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 4140 x e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 900 x e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 945 e^{x^{3}} \tan^{11}{\left(\frac{x}{2} \right)} + 3045 e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 630 e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + \frac{7251 e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)}}{2} + 1080 e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{3831 e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)}}{2} + 450 e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + \frac{825 e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)}}{2} + \frac{45 e^{x^{3}} \tan{\left(\frac{x}{2} \right)}}{2} - 945 \tan^{11}{\left(\frac{x}{2} \right)} - 3045 \tan^{9}{\left(\frac{x}{2} \right)} - \frac{7251 \tan^{7}{\left(\frac{x}{2} \right)}}{2} - \frac{3831 \tan^{5}{\left(\frac{x}{2} \right)}}{2} - \frac{825 \tan^{3}{\left(\frac{x}{2} \right)}}{2} - \frac{45 \tan{\left(\frac{x}{2} \right)}}{2}\right)}{45 \tan{\left(\frac{x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(243 x^{10} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 1215 x^{8} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 1215 x^{8} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 1620 x^{7} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 2835 x^{6} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 4860 x^{6} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 2025 x^{6} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 4860 x^{5} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 4860 x^{5} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 3780 x^{4} e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 8640 x^{4} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 2160 x^{4} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 6210 x^{4} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 1350 x^{4} e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 5670 x^{3} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 9720 x^{3} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 4050 x^{3} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 2835 x^{2} e^{x^{3}} \tan^{10}{\left(\frac{x}{2} \right)} + 7875 x^{2} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 2700 x^{2} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + \frac{15255 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)}}{2} + 2700 x^{2} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 2925 x^{2} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + \frac{675 x^{2} e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)}}{2} + 2520 x e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 5760 x e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 360 x e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 4140 x e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 900 x e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 945 e^{x^{3}} \tan^{11}{\left(\frac{x}{2} \right)} + 3045 e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 630 e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + \frac{7251 e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)}}{2} + 1080 e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{3831 e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)}}{2} + 450 e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + \frac{825 e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)}}{2} + \frac{45 e^{x^{3}} \tan{\left(\frac{x}{2} \right)}}{2} - 945 \tan^{11}{\left(\frac{x}{2} \right)} - 3045 \tan^{9}{\left(\frac{x}{2} \right)} - \frac{7251 \tan^{7}{\left(\frac{x}{2} \right)}}{2} - \frac{3831 \tan^{5}{\left(\frac{x}{2} \right)}}{2} - \frac{825 \tan^{3}{\left(\frac{x}{2} \right)}}{2} - \frac{45 \tan{\left(\frac{x}{2} \right)}}{2}\right)}{\frac{d}{d x} \frac{45 \tan{\left(\frac{x}{2} \right)}}{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{729 x^{12} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 4374 x^{10} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 4374 x^{10} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 7290 x^{9} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{25515 x^{8} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)}}{2} + 21870 x^{8} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{18225 x^{8} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)}}{2} + 29160 x^{7} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 29160 x^{7} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 22680 x^{6} e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 51840 x^{6} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 17820 x^{6} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 37260 x^{6} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 8100 x^{6} e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 51030 x^{5} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 87480 x^{5} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 36450 x^{5} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 25515 x^{4} e^{x^{3}} \tan^{10}{\left(\frac{x}{2} \right)} + 70875 x^{4} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 38880 x^{4} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + \frac{137295 x^{4} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)}}{2} + 38880 x^{4} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 26325 x^{4} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + \frac{6075 x^{4} e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)}}{2} + 45360 x^{3} e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 103680 x^{3} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 9720 x^{3} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 74520 x^{3} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 16200 x^{3} e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 17010 x^{2} e^{x^{3}} \tan^{11}{\left(\frac{x}{2} \right)} + 54810 x^{2} e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 28350 x^{2} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 65259 x^{2} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 48600 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 34479 x^{2} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 20250 x^{2} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 7425 x^{2} e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 405 x^{2} e^{x^{3}} \tan{\left(\frac{x}{2} \right)} + 17010 x e^{x^{3}} \tan^{10}{\left(\frac{x}{2} \right)} + 47250 x e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 6480 x e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 45765 x e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 6480 x e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 17550 x e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 2025 x e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)} + \frac{10395 e^{x^{3}} \tan^{12}{\left(\frac{x}{2} \right)}}{2} + 18900 e^{x^{3}} \tan^{10}{\left(\frac{x}{2} \right)} + 5040 e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + \frac{105567 e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)}}{4} + 11520 e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 17838 e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 8280 e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + \frac{10815 e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)}}{2} + 1800 e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 630 e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)} + \frac{45 e^{x^{3}}}{4} - \frac{10395 \tan^{12}{\left(\frac{x}{2} \right)}}{2} - 18900 \tan^{10}{\left(\frac{x}{2} \right)} - \frac{105567 \tan^{8}{\left(\frac{x}{2} \right)}}{4} - 17478 \tan^{6}{\left(\frac{x}{2} \right)} - \frac{10815 \tan^{4}{\left(\frac{x}{2} \right)}}{2} - 630 \tan^{2}{\left(\frac{x}{2} \right)} - \frac{45}{4}}{\frac{45 \tan^{2}{\left(\frac{x}{2} \right)}}{4} + \frac{45}{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{729 x^{12} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 4374 x^{10} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 4374 x^{10} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 7290 x^{9} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{25515 x^{8} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)}}{2} + 21870 x^{8} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{18225 x^{8} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)}}{2} + 29160 x^{7} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 29160 x^{7} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 22680 x^{6} e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 51840 x^{6} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 17820 x^{6} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 37260 x^{6} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 8100 x^{6} e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 51030 x^{5} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 87480 x^{5} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 36450 x^{5} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 25515 x^{4} e^{x^{3}} \tan^{10}{\left(\frac{x}{2} \right)} + 70875 x^{4} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 38880 x^{4} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + \frac{137295 x^{4} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)}}{2} + 38880 x^{4} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 26325 x^{4} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + \frac{6075 x^{4} e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)}}{2} + 45360 x^{3} e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 103680 x^{3} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 9720 x^{3} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 74520 x^{3} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 16200 x^{3} e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 17010 x^{2} e^{x^{3}} \tan^{11}{\left(\frac{x}{2} \right)} + 54810 x^{2} e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + 28350 x^{2} e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 65259 x^{2} e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 48600 x^{2} e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 34479 x^{2} e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 20250 x^{2} e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 7425 x^{2} e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 405 x^{2} e^{x^{3}} \tan{\left(\frac{x}{2} \right)} + 17010 x e^{x^{3}} \tan^{10}{\left(\frac{x}{2} \right)} + 47250 x e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)} + 6480 x e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 45765 x e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 6480 x e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + 17550 x e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)} + 2025 x e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)} + \frac{10395 e^{x^{3}} \tan^{12}{\left(\frac{x}{2} \right)}}{2} + 18900 e^{x^{3}} \tan^{10}{\left(\frac{x}{2} \right)} + 5040 e^{x^{3}} \tan^{9}{\left(\frac{x}{2} \right)} + \frac{105567 e^{x^{3}} \tan^{8}{\left(\frac{x}{2} \right)}}{4} + 11520 e^{x^{3}} \tan^{7}{\left(\frac{x}{2} \right)} + 17838 e^{x^{3}} \tan^{6}{\left(\frac{x}{2} \right)} + 8280 e^{x^{3}} \tan^{5}{\left(\frac{x}{2} \right)} + \frac{10815 e^{x^{3}} \tan^{4}{\left(\frac{x}{2} \right)}}{2} + 1800 e^{x^{3}} \tan^{3}{\left(\frac{x}{2} \right)} + 630 e^{x^{3}} \tan^{2}{\left(\frac{x}{2} \right)} + \frac{45 e^{x^{3}}}{4} - \frac{10395 \tan^{12}{\left(\frac{x}{2} \right)}}{2} - 18900 \tan^{10}{\left(\frac{x}{2} \right)} - \frac{105567 \tan^{8}{\left(\frac{x}{2} \right)}}{4} - 17478 \tan^{6}{\left(\frac{x}{2} \right)} - \frac{10815 \tan^{4}{\left(\frac{x}{2} \right)}}{2} - 630 \tan^{2}{\left(\frac{x}{2} \right)} - \frac{45}{4}}{\frac{45 \tan^{2}{\left(\frac{x}{2} \right)}}{4} + \frac{45}{4}}\right)$$
=
$$-\infty$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 6 vez (veces)
Gráfica
A la izquierda y a la derecha
[src]
/ / 3\ 3 \
| \x / x |
lim |-1 + E - -------|
x->0+| 6/x\|
| tan |-||
\ \2//
$$\lim_{x \to 0^+}\left(- \frac{x^{3}}{\tan^{6}{\left(\frac{x}{2} \right)}} + \left(e^{x^{3}} - 1\right)\right)$$
-oo
$$-\infty$$
= -220344032.040618
/ / 3\ 3 \
| \x / x |
lim |-1 + E - -------|
x->0-| 6/x\|
| tan |-||
\ \2//
$$\lim_{x \to 0^-}\left(- \frac{x^{3}}{\tan^{6}{\left(\frac{x}{2} \right)}} + \left(e^{x^{3}} - 1\right)\right)$$
oo
$$\infty$$
= 220344032.040618
= 220344032.040618
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(- \frac{x^{3}}{\tan^{6}{\left(\frac{x}{2} \right)}} + \left(e^{x^{3}} - 1\right)\right) = -\infty$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(- \frac{x^{3}}{\tan^{6}{\left(\frac{x}{2} \right)}} + \left(e^{x^{3}} - 1\right)\right) = -\infty$$
$$\lim_{x \to \infty}\left(- \frac{x^{3}}{\tan^{6}{\left(\frac{x}{2} \right)}} + \left(e^{x^{3}} - 1\right)\right) = \infty$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(- \frac{x^{3}}{\tan^{6}{\left(\frac{x}{2} \right)}} + \left(e^{x^{3}} - 1\right)\right) = \frac{-1 - \tan^{6}{\left(\frac{1}{2} \right)} + e \tan^{6}{\left(\frac{1}{2} \right)}}{\tan^{6}{\left(\frac{1}{2} \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(- \frac{x^{3}}{\tan^{6}{\left(\frac{x}{2} \right)}} + \left(e^{x^{3}} - 1\right)\right) = \frac{-1 - \tan^{6}{\left(\frac{1}{2} \right)} + e \tan^{6}{\left(\frac{1}{2} \right)}}{\tan^{6}{\left(\frac{1}{2} \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(- \frac{x^{3}}{\tan^{6}{\left(\frac{x}{2} \right)}} + \left(e^{x^{3}} - 1\right)\right)$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(- \frac{x^{3}}{\tan^{6}{\left(\frac{x}{2} \right)}} + \left(e^{x^{3}} - 1\right)\right) = -\infty$$
$$\lim_{x \to \infty}\left(- \frac{x^{3}}{\tan^{6}{\left(\frac{x}{2} \right)}} + \left(e^{x^{3}} - 1\right)\right) = \infty$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(- \frac{x^{3}}{\tan^{6}{\left(\frac{x}{2} \right)}} + \left(e^{x^{3}} - 1\right)\right) = \frac{-1 - \tan^{6}{\left(\frac{1}{2} \right)} + e \tan^{6}{\left(\frac{1}{2} \right)}}{\tan^{6}{\left(\frac{1}{2} \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(- \frac{x^{3}}{\tan^{6}{\left(\frac{x}{2} \right)}} + \left(e^{x^{3}} - 1\right)\right) = \frac{-1 - \tan^{6}{\left(\frac{1}{2} \right)} + e \tan^{6}{\left(\frac{1}{2} \right)}}{\tan^{6}{\left(\frac{1}{2} \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(- \frac{x^{3}}{\tan^{6}{\left(\frac{x}{2} \right)}} + \left(e^{x^{3}} - 1\right)\right)$$
Más detalles con x→-oo