Sr Examen
Otras calculadoras:
-
¿Cómo usar?
- Límite de la función:
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Límite de (-3+x^2+2*x)/(x^3+3*x+4*x^2)
- Límite de -sin(a/2-x/2)*tan(pi*x/(2*a))
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Límite de (sqrt(17+3*x)-sqrt(12+2*x))/(15+x^2+8*x)
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Límite de (-2+sqrt(4+x^2))/(-3+sqrt(9+x^2))
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Expresiones idénticas
- atan(x^ dos)^ dos /log(e^(- dos *x^ dos)+sin(dos *x^ dos))
- arco tangente de gente de (x al cuadrado ) al cuadrado dividir por logaritmo de (e en el grado ( menos 2 multiplicar por x al cuadrado ) más seno de (2 multiplicar por x al cuadrado ))
- arco tangente de gente de (x en el grado dos) en el grado dos dividir por logaritmo de (e en el grado ( menos dos multiplicar por x en el grado dos) más seno de (dos multiplicar por x en el grado dos))
- atan(x2)2/log(e(-2*x2)+sin(2*x2))
- atanx22/loge-2*x2+sin2*x2
- atan(x²)²/log(e^(-2*x²)+sin(2*x²))
- atan(x en el grado 2) en el grado 2/log(e en el grado (-2*x en el grado 2)+sin(2*x en el grado 2))
- atan(x^2)^2/log(e^(-2x^2)+sin(2x^2))
- atan(x2)2/log(e(-2x2)+sin(2x2))
- atanx22/loge-2x2+sin2x2
- atanx^2^2/loge^-2x^2+sin2x^2
- atan(x^2)^2 dividir por log(e^(-2*x^2)+sin(2*x^2))
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Expresiones semejantes
- atan(x^2)^2/log(e^(2*x^2)+sin(2*x^2))
- atan(x^2)^2/log(e^(-2*x^2)-sin(2*x^2))
- arctan(x^2)^2/log(e^(-2*x^2)+sin(2*x^2))
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Expresiones con funciones
- Arcotangente arctan
- atan(2*x/(-1+2*x))
- atan(sqrt(3)*sqrt(x))/log((-1+x)^2)
- atan(x^2-3*x)/(1-cos(-3+x))
- atan(n/2)
- atan(3*x)*sin(3*x)
- Logaritmo log
- log(1+x)/2+log(-1+x)/2-log(1+x^2-x)/2
- log(x)/(2+sqrt(x))
- log(-62+x^2-2*x)/((-27+x^2-6*x)*log(e))
- log(x*sin(m))/(log(x)*sin(x))
- log(2-x/a)^tan(pi*x/(2*a))
- Seno sin
- sin(k*x)/(k*x)
- sin(x*3^(-n))^(1/n)
- sin(cos(-18+x))/sin(sin(-18+x))
- sin(3*x)^2/(x*tan(x)^2)
- sin(3*x)^2*tan(3*x)/(6*x^3)
Límite de la función atan(x^2)^2/log(e^(-2*x^2)+sin(2*x^2))
Solución
Ha introducido
[src]
/ 2/ 2\ \
| atan \x / |
lim |-----------------------|
x->0+| / 2 \|
| | -2*x / 2\||
\log\E + sin\2*x ///
$$\lim_{x \to 0^+}\left(\frac{\operatorname{atan}^{2}{\left(x^{2} \right)}}{\log{\left(\sin{\left(2 x^{2} \right)} + e^{- 2 x^{2}} \right)}}\right)$$
Limit(atan(x^2)^2/log(E^(-2*x^2) + sin(2*x^2)), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+} \operatorname{atan}^{2}{\left(x^{2} \right)} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \log{\left(\left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} \right)} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{\operatorname{atan}^{2}{\left(x^{2} \right)}}{\log{\left(\sin{\left(2 x^{2} \right)} + e^{- 2 x^{2}} \right)}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\frac{\operatorname{atan}^{2}{\left(x^{2} \right)}}{\log{\left(\left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \operatorname{atan}^{2}{\left(x^{2} \right)}}{\frac{d}{d x} \log{\left(\left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{4 x \left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} \operatorname{atan}{\left(x^{2} \right)}}{\left(x^{4} + 1\right) \left(- 4 x \left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} + \left(4 x e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 4 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{4 x \operatorname{atan}{\left(x^{2} \right)}}{- 4 x \left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} + \left(4 x e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 4 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} 4 x \operatorname{atan}{\left(x^{2} \right)}}{\frac{d}{d x} \left(- 4 x \left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} + \left(4 x e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 4 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{8 x^{2}}{x^{4} + 1} + 4 \operatorname{atan}{\left(x^{2} \right)}}{16 x^{2} \left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} - 8 x \left(4 x e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 4 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}} + \left(- 4 e^{2 x^{2}} \sin{\left(2 x^{2} \right)} - 4\right) e^{- 2 x^{2}} + \left(32 x^{2} e^{2 x^{2}} \cos{\left(2 x^{2} \right)} + 4 e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 4 e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\frac{8 x^{2}}{x^{4} + 1} + 4 \operatorname{atan}{\left(x^{2} \right)}\right)}{\frac{d}{d x} \left(16 x^{2} \left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} - 8 x \left(4 x e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 4 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}} + \left(- 4 e^{2 x^{2}} \sin{\left(2 x^{2} \right)} - 4\right) e^{- 2 x^{2}} + \left(32 x^{2} e^{2 x^{2}} \cos{\left(2 x^{2} \right)} + 4 e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 4 e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{32 x^{5}}{\left(x^{4} + 1\right)^{2}} + \frac{24 x}{x^{4} + 1}}{- 64 x^{3} \left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} + 48 x^{2} \left(4 x e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 4 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}} - 4 x \left(- 4 e^{2 x^{2}} \sin{\left(2 x^{2} \right)} - 4\right) e^{- 2 x^{2}} + 32 x \left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} - 12 x \left(32 x^{2} e^{2 x^{2}} \cos{\left(2 x^{2} \right)} + 4 e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 4 e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}} + \left(- 32 x e^{2 x^{2}} \sin{\left(2 x^{2} \right)} - 32 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}} + \left(- 16 x e^{2 x^{2}} \sin{\left(2 x^{2} \right)} - 16 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}} + \left(- 128 x^{3} e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 128 x^{3} e^{2 x^{2}} \cos{\left(2 x^{2} \right)} + 96 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{32 x^{5}}{\left(x^{4} + 1\right)^{2}} + \frac{24 x}{x^{4} + 1}\right)}{\frac{d}{d x} \left(- 64 x^{3} \left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} + 48 x^{2} \left(4 x e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 4 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}} - 4 x \left(- 4 e^{2 x^{2}} \sin{\left(2 x^{2} \right)} - 4\right) e^{- 2 x^{2}} + 32 x \left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} - 12 x \left(32 x^{2} e^{2 x^{2}} \cos{\left(2 x^{2} \right)} + 4 e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 4 e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}} + \left(- 32 x e^{2 x^{2}} \sin{\left(2 x^{2} \right)} - 32 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}} + \left(- 16 x e^{2 x^{2}} \sin{\left(2 x^{2} \right)} - 16 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}} + \left(- 128 x^{3} e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 128 x^{3} e^{2 x^{2}} \cos{\left(2 x^{2} \right)} + 96 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{256 x^{8}}{x^{12} + 3 x^{8} + 3 x^{4} + 1} - \frac{256 x^{4}}{x^{8} + 2 x^{4} + 1} + \frac{24}{x^{4} + 1}}{256 x^{4} \sin{\left(2 x^{2} \right)} + 256 x^{4} e^{- 2 x^{2}} - 384 x^{2} \cos{\left(2 x^{2} \right)} - 384 x^{2} e^{- 2 x^{2}} - 48 \sin{\left(2 x^{2} \right)} + 48 e^{- 2 x^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{256 x^{8}}{x^{12} + 3 x^{8} + 3 x^{4} + 1} - \frac{256 x^{4}}{x^{8} + 2 x^{4} + 1} + \frac{24}{x^{4} + 1}}{256 x^{4} \sin{\left(2 x^{2} \right)} + 256 x^{4} e^{- 2 x^{2}} - 384 x^{2} \cos{\left(2 x^{2} \right)} - 384 x^{2} e^{- 2 x^{2}} - 48 \sin{\left(2 x^{2} \right)} + 48 e^{- 2 x^{2}}}\right)$$
=
$$\frac{1}{2}$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 4 vez (veces)
0/0,
tal que el límite para el numerador es
$$\lim_{x \to 0^+} \operatorname{atan}^{2}{\left(x^{2} \right)} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \log{\left(\left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} \right)} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{\operatorname{atan}^{2}{\left(x^{2} \right)}}{\log{\left(\sin{\left(2 x^{2} \right)} + e^{- 2 x^{2}} \right)}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\frac{\operatorname{atan}^{2}{\left(x^{2} \right)}}{\log{\left(\left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \operatorname{atan}^{2}{\left(x^{2} \right)}}{\frac{d}{d x} \log{\left(\left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{4 x \left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} \operatorname{atan}{\left(x^{2} \right)}}{\left(x^{4} + 1\right) \left(- 4 x \left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} + \left(4 x e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 4 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{4 x \operatorname{atan}{\left(x^{2} \right)}}{- 4 x \left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} + \left(4 x e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 4 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} 4 x \operatorname{atan}{\left(x^{2} \right)}}{\frac{d}{d x} \left(- 4 x \left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} + \left(4 x e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 4 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{8 x^{2}}{x^{4} + 1} + 4 \operatorname{atan}{\left(x^{2} \right)}}{16 x^{2} \left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} - 8 x \left(4 x e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 4 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}} + \left(- 4 e^{2 x^{2}} \sin{\left(2 x^{2} \right)} - 4\right) e^{- 2 x^{2}} + \left(32 x^{2} e^{2 x^{2}} \cos{\left(2 x^{2} \right)} + 4 e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 4 e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\frac{8 x^{2}}{x^{4} + 1} + 4 \operatorname{atan}{\left(x^{2} \right)}\right)}{\frac{d}{d x} \left(16 x^{2} \left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} - 8 x \left(4 x e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 4 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}} + \left(- 4 e^{2 x^{2}} \sin{\left(2 x^{2} \right)} - 4\right) e^{- 2 x^{2}} + \left(32 x^{2} e^{2 x^{2}} \cos{\left(2 x^{2} \right)} + 4 e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 4 e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{32 x^{5}}{\left(x^{4} + 1\right)^{2}} + \frac{24 x}{x^{4} + 1}}{- 64 x^{3} \left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} + 48 x^{2} \left(4 x e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 4 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}} - 4 x \left(- 4 e^{2 x^{2}} \sin{\left(2 x^{2} \right)} - 4\right) e^{- 2 x^{2}} + 32 x \left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} - 12 x \left(32 x^{2} e^{2 x^{2}} \cos{\left(2 x^{2} \right)} + 4 e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 4 e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}} + \left(- 32 x e^{2 x^{2}} \sin{\left(2 x^{2} \right)} - 32 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}} + \left(- 16 x e^{2 x^{2}} \sin{\left(2 x^{2} \right)} - 16 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}} + \left(- 128 x^{3} e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 128 x^{3} e^{2 x^{2}} \cos{\left(2 x^{2} \right)} + 96 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{32 x^{5}}{\left(x^{4} + 1\right)^{2}} + \frac{24 x}{x^{4} + 1}\right)}{\frac{d}{d x} \left(- 64 x^{3} \left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} + 48 x^{2} \left(4 x e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 4 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}} - 4 x \left(- 4 e^{2 x^{2}} \sin{\left(2 x^{2} \right)} - 4\right) e^{- 2 x^{2}} + 32 x \left(e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 1\right) e^{- 2 x^{2}} - 12 x \left(32 x^{2} e^{2 x^{2}} \cos{\left(2 x^{2} \right)} + 4 e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 4 e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}} + \left(- 32 x e^{2 x^{2}} \sin{\left(2 x^{2} \right)} - 32 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}} + \left(- 16 x e^{2 x^{2}} \sin{\left(2 x^{2} \right)} - 16 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}} + \left(- 128 x^{3} e^{2 x^{2}} \sin{\left(2 x^{2} \right)} + 128 x^{3} e^{2 x^{2}} \cos{\left(2 x^{2} \right)} + 96 x e^{2 x^{2}} \cos{\left(2 x^{2} \right)}\right) e^{- 2 x^{2}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{256 x^{8}}{x^{12} + 3 x^{8} + 3 x^{4} + 1} - \frac{256 x^{4}}{x^{8} + 2 x^{4} + 1} + \frac{24}{x^{4} + 1}}{256 x^{4} \sin{\left(2 x^{2} \right)} + 256 x^{4} e^{- 2 x^{2}} - 384 x^{2} \cos{\left(2 x^{2} \right)} - 384 x^{2} e^{- 2 x^{2}} - 48 \sin{\left(2 x^{2} \right)} + 48 e^{- 2 x^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{256 x^{8}}{x^{12} + 3 x^{8} + 3 x^{4} + 1} - \frac{256 x^{4}}{x^{8} + 2 x^{4} + 1} + \frac{24}{x^{4} + 1}}{256 x^{4} \sin{\left(2 x^{2} \right)} + 256 x^{4} e^{- 2 x^{2}} - 384 x^{2} \cos{\left(2 x^{2} \right)} - 384 x^{2} e^{- 2 x^{2}} - 48 \sin{\left(2 x^{2} \right)} + 48 e^{- 2 x^{2}}}\right)$$
=
$$\frac{1}{2}$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 4 vez (veces)
Gráfica
A la izquierda y a la derecha
[src]
/ 2/ 2\ \
| atan \x / |
lim |-----------------------|
x->0+| / 2 \|
| | -2*x / 2\||
\log\E + sin\2*x ///
$$\lim_{x \to 0^+}\left(\frac{\operatorname{atan}^{2}{\left(x^{2} \right)}}{\log{\left(\sin{\left(2 x^{2} \right)} + e^{- 2 x^{2}} \right)}}\right)$$
1/2
$$\frac{1}{2}$$
= 0.5
/ 2/ 2\ \
| atan \x / |
lim |-----------------------|
x->0-| / 2 \|
| | -2*x / 2\||
\log\E + sin\2*x ///
$$\lim_{x \to 0^-}\left(\frac{\operatorname{atan}^{2}{\left(x^{2} \right)}}{\log{\left(\sin{\left(2 x^{2} \right)} + e^{- 2 x^{2}} \right)}}\right)$$
1/2
$$\frac{1}{2}$$
= 0.5
= 0.5
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\operatorname{atan}^{2}{\left(x^{2} \right)}}{\log{\left(\sin{\left(2 x^{2} \right)} + e^{- 2 x^{2}} \right)}}\right) = \frac{1}{2}$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\operatorname{atan}^{2}{\left(x^{2} \right)}}{\log{\left(\sin{\left(2 x^{2} \right)} + e^{- 2 x^{2}} \right)}}\right) = \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{\operatorname{atan}^{2}{\left(x^{2} \right)}}{\log{\left(\sin{\left(2 x^{2} \right)} + e^{- 2 x^{2}} \right)}}\right) = \frac{\pi^{2}}{4 \log{\left(\left\langle -1, 1\right\rangle \right)}}$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\operatorname{atan}^{2}{\left(x^{2} \right)}}{\log{\left(\sin{\left(2 x^{2} \right)} + e^{- 2 x^{2}} \right)}}\right) = \frac{\pi^{2}}{16 \log{\left(e^{-2} + \sin{\left(2 \right)} \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\operatorname{atan}^{2}{\left(x^{2} \right)}}{\log{\left(\sin{\left(2 x^{2} \right)} + e^{- 2 x^{2}} \right)}}\right) = \frac{\pi^{2}}{16 \log{\left(e^{-2} + \sin{\left(2 \right)} \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\operatorname{atan}^{2}{\left(x^{2} \right)}}{\log{\left(\sin{\left(2 x^{2} \right)} + e^{- 2 x^{2}} \right)}}\right) = \frac{\pi^{2}}{4 \log{\left(\left\langle -1, 1\right\rangle \right)}}$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\operatorname{atan}^{2}{\left(x^{2} \right)}}{\log{\left(\sin{\left(2 x^{2} \right)} + e^{- 2 x^{2}} \right)}}\right) = \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{\operatorname{atan}^{2}{\left(x^{2} \right)}}{\log{\left(\sin{\left(2 x^{2} \right)} + e^{- 2 x^{2}} \right)}}\right) = \frac{\pi^{2}}{4 \log{\left(\left\langle -1, 1\right\rangle \right)}}$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\operatorname{atan}^{2}{\left(x^{2} \right)}}{\log{\left(\sin{\left(2 x^{2} \right)} + e^{- 2 x^{2}} \right)}}\right) = \frac{\pi^{2}}{16 \log{\left(e^{-2} + \sin{\left(2 \right)} \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\operatorname{atan}^{2}{\left(x^{2} \right)}}{\log{\left(\sin{\left(2 x^{2} \right)} + e^{- 2 x^{2}} \right)}}\right) = \frac{\pi^{2}}{16 \log{\left(e^{-2} + \sin{\left(2 \right)} \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\operatorname{atan}^{2}{\left(x^{2} \right)}}{\log{\left(\sin{\left(2 x^{2} \right)} + e^{- 2 x^{2}} \right)}}\right) = \frac{\pi^{2}}{4 \log{\left(\left\langle -1, 1\right\rangle \right)}}$$
Más detalles con x→-oo