Sr Examen
Otras calculadoras:
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- Límite de la función:
-
Límite de (-9+x^2)/(3+x)
-
Límite de x^2/(1-cos(6*x))
-
Límite de (4+x^2-5*x)/(8+x^2-6*x)
-
Límite de (-3+x^2-2*x)/(-9+x^2)
-
Expresiones idénticas
- (-Abs(- uno +sin(dos *x))+Abs(uno +sin(dos *x)))/tan(x)
- ( menos Abs( menos 1 más seno de (2 multiplicar por x)) más Abs(1 más seno de (2 multiplicar por x))) dividir por tangente de (x)
- ( menos Abs( menos uno más seno de (dos multiplicar por x)) más Abs(uno más seno de (dos multiplicar por x))) dividir por tangente de (x)
- (-Abs(-1+sin(2x))+Abs(1+sin(2x)))/tan(x)
- -Abs-1+sin2x+Abs1+sin2x/tanx
- (-Abs(-1+sin(2*x))+Abs(1+sin(2*x))) dividir por tan(x)
-
Expresiones semejantes
- (-Abs(1+sin(2*x))+Abs(1+sin(2*x)))/tan(x)
- (-Abs(-1+sin(2*x))-Abs(1+sin(2*x)))/tan(x)
- (Abs(-1+sin(2*x))+Abs(1+sin(2*x)))/tan(x)
- (-Abs(-1+sin(2*x))+Abs(1-sin(2*x)))/tan(x)
- (-Abs(-1-sin(2*x))+Abs(1+sin(2*x)))/tan(x)
-
Expresiones con funciones
- Seno sin
- sin(5*x)/sin(2*x)
- sin(3*x)/(3*x)
- sin(x^2)/x
- sin(x)/|x|
- sin(x)/asin(x)
- Seno sin
- sin(5*x)/sin(2*x)
- sin(3*x)/(3*x)
- sin(x^2)/x
- sin(x)/|x|
- sin(x)/asin(x)
- Tangente tan
- tan(3*x)/(2*x)
- tan(3*x)/sin(x)
- tan(2*x)/asin(x)
- tan(9*x)/x
- tan(x)/x^3
Límite de la función (-Abs(-1+sin(2*x))+Abs(1+sin(2*x)))/tan(x)
Solución
Ha introducido
[src]
/-|-1 + sin(2*x)| + |1 + sin(2*x)|\ lim |---------------------------------| x->0+\ tan(x) /
$$\lim_{x \to 0^+}\left(\frac{- \left|{\sin{\left(2 x \right)} - 1}\right| + \left|{\sin{\left(2 x \right)} + 1}\right|}{\tan{\left(x \right)}}\right)$$
Limit((-Abs(-1 + sin(2*x)) + Abs(1 + sin(2*x)))/tan(x), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(- \left|{\sin{\left(2 x \right)} - 1}\right| + \left|{\sin{\left(2 x \right)} + 1}\right|\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \tan{\left(x \right)} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{- \left|{\sin{\left(2 x \right)} - 1}\right| + \left|{\sin{\left(2 x \right)} + 1}\right|}{\tan{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \left|{\sin{\left(2 x \right)} - 1}\right| + \left|{\sin{\left(2 x \right)} + 1}\right|\right)}{\frac{d}{d x} \tan{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{2 \sin^{2}{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} + 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\sin{\left(2 x \right)} + 1} - \frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \cos{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} + 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\sin{\left(2 x \right)} + 1} + \frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \cos{\left(2 \operatorname{re}{\left(x\right)} \right)} \cosh^{2}{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} + 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\sin{\left(2 x \right)} + 1} + \frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} + 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\sin{\left(2 x \right)} + 1} + \frac{2 \cos^{2}{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} + 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\sin{\left(2 x \right)} + 1} + \frac{2 \cos{\left(2 \operatorname{re}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} + 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\sin{\left(2 x \right)} + 1} - \frac{2 \sin^{2}{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} - 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\sin{\left(2 x \right)} - 1} + \frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \cos{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} - 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\sin{\left(2 x \right)} - 1} - \frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \cos{\left(2 \operatorname{re}{\left(x\right)} \right)} \cosh^{2}{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} - 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\sin{\left(2 x \right)} - 1} + \frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} - 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\sin{\left(2 x \right)} - 1} - \frac{2 \cos^{2}{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} - 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\sin{\left(2 x \right)} - 1} + \frac{2 \cos{\left(2 \operatorname{re}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} - 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\sin{\left(2 x \right)} - 1}}{\tan^{2}{\left(x \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{2 \sin^{2}{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} + 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\sin{\left(2 x \right)} + 1} - \frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \cos{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} + 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\sin{\left(2 x \right)} + 1} + \frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \cos{\left(2 \operatorname{re}{\left(x\right)} \right)} \cosh^{2}{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} + 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\sin{\left(2 x \right)} + 1} + \frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} + 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\sin{\left(2 x \right)} + 1} + \frac{2 \cos^{2}{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} + 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\sin{\left(2 x \right)} + 1} + \frac{2 \cos{\left(2 \operatorname{re}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} + 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\sin{\left(2 x \right)} + 1} - \frac{2 \sin^{2}{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} - 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\sin{\left(2 x \right)} - 1} + \frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \cos{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} - 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\sin{\left(2 x \right)} - 1} - \frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \cos{\left(2 \operatorname{re}{\left(x\right)} \right)} \cosh^{2}{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} - 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\sin{\left(2 x \right)} - 1} + \frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} - 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\sin{\left(2 x \right)} - 1} - \frac{2 \cos^{2}{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} - 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\sin{\left(2 x \right)} - 1} + \frac{2 \cos{\left(2 \operatorname{re}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} - 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\sin{\left(2 x \right)} - 1}}{\tan^{2}{\left(x \right)} + 1}\right)$$
=
$$4$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 1 vez (veces)
0/0,
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(- \left|{\sin{\left(2 x \right)} - 1}\right| + \left|{\sin{\left(2 x \right)} + 1}\right|\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \tan{\left(x \right)} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{- \left|{\sin{\left(2 x \right)} - 1}\right| + \left|{\sin{\left(2 x \right)} + 1}\right|}{\tan{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \left|{\sin{\left(2 x \right)} - 1}\right| + \left|{\sin{\left(2 x \right)} + 1}\right|\right)}{\frac{d}{d x} \tan{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{2 \sin^{2}{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} + 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\sin{\left(2 x \right)} + 1} - \frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \cos{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} + 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\sin{\left(2 x \right)} + 1} + \frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \cos{\left(2 \operatorname{re}{\left(x\right)} \right)} \cosh^{2}{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} + 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\sin{\left(2 x \right)} + 1} + \frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} + 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\sin{\left(2 x \right)} + 1} + \frac{2 \cos^{2}{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} + 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\sin{\left(2 x \right)} + 1} + \frac{2 \cos{\left(2 \operatorname{re}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} + 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\sin{\left(2 x \right)} + 1} - \frac{2 \sin^{2}{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} - 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\sin{\left(2 x \right)} - 1} + \frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \cos{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} - 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\sin{\left(2 x \right)} - 1} - \frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \cos{\left(2 \operatorname{re}{\left(x\right)} \right)} \cosh^{2}{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} - 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\sin{\left(2 x \right)} - 1} + \frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} - 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\sin{\left(2 x \right)} - 1} - \frac{2 \cos^{2}{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} - 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\sin{\left(2 x \right)} - 1} + \frac{2 \cos{\left(2 \operatorname{re}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} - 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\sin{\left(2 x \right)} - 1}}{\tan^{2}{\left(x \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{2 \sin^{2}{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} + 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\sin{\left(2 x \right)} + 1} - \frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \cos{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} + 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\sin{\left(2 x \right)} + 1} + \frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \cos{\left(2 \operatorname{re}{\left(x\right)} \right)} \cosh^{2}{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} + 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\sin{\left(2 x \right)} + 1} + \frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} + 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\sin{\left(2 x \right)} + 1} + \frac{2 \cos^{2}{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} + 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\sin{\left(2 x \right)} + 1} + \frac{2 \cos{\left(2 \operatorname{re}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} + 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\sin{\left(2 x \right)} + 1} - \frac{2 \sin^{2}{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} - 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\sin{\left(2 x \right)} - 1} + \frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \cos{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} - 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\sin{\left(2 x \right)} - 1} - \frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \cos{\left(2 \operatorname{re}{\left(x\right)} \right)} \cosh^{2}{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} - 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\sin{\left(2 x \right)} - 1} + \frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} - 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\sin{\left(2 x \right)} - 1} - \frac{2 \cos^{2}{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} - 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\sin{\left(2 x \right)} - 1} + \frac{2 \cos{\left(2 \operatorname{re}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} \operatorname{sign}{\left(\sin{\left(2 x \right)} - 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\sin{\left(2 x \right)} - 1}}{\tan^{2}{\left(x \right)} + 1}\right)$$
=
$$4$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 1 vez (veces)
Gráfica
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{- \left|{\sin{\left(2 x \right)} - 1}\right| + \left|{\sin{\left(2 x \right)} + 1}\right|}{\tan{\left(x \right)}}\right) = 4$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{- \left|{\sin{\left(2 x \right)} - 1}\right| + \left|{\sin{\left(2 x \right)} + 1}\right|}{\tan{\left(x \right)}}\right) = 4$$
$$\lim_{x \to \infty}\left(\frac{- \left|{\sin{\left(2 x \right)} - 1}\right| + \left|{\sin{\left(2 x \right)} + 1}\right|}{\tan{\left(x \right)}}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{- \left|{\sin{\left(2 x \right)} - 1}\right| + \left|{\sin{\left(2 x \right)} + 1}\right|}{\tan{\left(x \right)}}\right) = \frac{2 \sin{\left(2 \right)}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{- \left|{\sin{\left(2 x \right)} - 1}\right| + \left|{\sin{\left(2 x \right)} + 1}\right|}{\tan{\left(x \right)}}\right) = \frac{2 \sin{\left(2 \right)}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{- \left|{\sin{\left(2 x \right)} - 1}\right| + \left|{\sin{\left(2 x \right)} + 1}\right|}{\tan{\left(x \right)}}\right)$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{- \left|{\sin{\left(2 x \right)} - 1}\right| + \left|{\sin{\left(2 x \right)} + 1}\right|}{\tan{\left(x \right)}}\right) = 4$$
$$\lim_{x \to \infty}\left(\frac{- \left|{\sin{\left(2 x \right)} - 1}\right| + \left|{\sin{\left(2 x \right)} + 1}\right|}{\tan{\left(x \right)}}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{- \left|{\sin{\left(2 x \right)} - 1}\right| + \left|{\sin{\left(2 x \right)} + 1}\right|}{\tan{\left(x \right)}}\right) = \frac{2 \sin{\left(2 \right)}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{- \left|{\sin{\left(2 x \right)} - 1}\right| + \left|{\sin{\left(2 x \right)} + 1}\right|}{\tan{\left(x \right)}}\right) = \frac{2 \sin{\left(2 \right)}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{- \left|{\sin{\left(2 x \right)} - 1}\right| + \left|{\sin{\left(2 x \right)} + 1}\right|}{\tan{\left(x \right)}}\right)$$
Más detalles con x→-oo
A la izquierda y a la derecha
[src]
/-|-1 + sin(2*x)| + |1 + sin(2*x)|\ lim |---------------------------------| x->0+\ tan(x) /
$$\lim_{x \to 0^+}\left(\frac{- \left|{\sin{\left(2 x \right)} - 1}\right| + \left|{\sin{\left(2 x \right)} + 1}\right|}{\tan{\left(x \right)}}\right)$$
4
$$4$$
= 4
/-|-1 + sin(2*x)| + |1 + sin(2*x)|\ lim |---------------------------------| x->0-\ tan(x) /
$$\lim_{x \to 0^-}\left(\frac{- \left|{\sin{\left(2 x \right)} - 1}\right| + \left|{\sin{\left(2 x \right)} + 1}\right|}{\tan{\left(x \right)}}\right)$$
4
$$4$$
= 4
= 4