Sr Examen
Otras calculadoras:
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- Límite de la función:
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Límite de (1-cos(5*x))/x^2
-
Límite de x/(-1+sqrt(1+3*x))
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Límite de (-27+x^3)/(-9+x^2)
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Límite de (-1-4*x+5*x^2)/(-1+x)
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Expresiones idénticas
- cot(x)^ dos *sin(x)^ dos *(-sin(x)^(- dos *x)/cot(x)^ dos +cos(x))
- cotangente de (x) al cuadrado multiplicar por seno de (x) al cuadrado multiplicar por ( menos seno de (x) en el grado ( menos 2 multiplicar por x) dividir por cotangente de (x) al cuadrado más coseno de (x))
- cotangente de (x) en el grado dos multiplicar por seno de (x) en el grado dos multiplicar por ( menos seno de (x) en el grado ( menos dos multiplicar por x) dividir por cotangente de (x) en el grado dos más coseno de (x))
- cot(x)2*sin(x)2*(-sin(x)(-2*x)/cot(x)2+cos(x))
- cotx2*sinx2*-sinx-2*x/cotx2+cosx
- cot(x)²*sin(x)²*(-sin(x)^(-2*x)/cot(x)²+cos(x))
- cot(x) en el grado 2*sin(x) en el grado 2*(-sin(x) en el grado (-2*x)/cot(x) en el grado 2+cos(x))
- cot(x)^2sin(x)^2(-sin(x)^(-2x)/cot(x)^2+cos(x))
- cot(x)2sin(x)2(-sin(x)(-2x)/cot(x)2+cos(x))
- cotx2sinx2-sinx-2x/cotx2+cosx
- cotx^2sinx^2-sinx^-2x/cotx^2+cosx
- cot(x)^2*sin(x)^2*(-sin(x)^(-2*x) dividir por cot(x)^2+cos(x))
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Expresiones semejantes
- cot(x)^2*sin(x)^2*(-sin(x)^(-2*x)/cot(x)^2-cos(x))
- cot(x)^2*sin(x)^2*(sin(x)^(-2*x)/cot(x)^2+cos(x))
- cot(x)^2*sin(x)^2*(-sin(x)^(2*x)/cot(x)^2+cos(x))
- cot(x)^2*sinx^2*(-sinx^(-2*x)/cot(x)^2+cosx)
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Expresiones con funciones
- Cotangente cot
- cot(2*x)*sin(5*x)
- cot(3*x)*sin(6*x)
- cot(x)*tan(x)
- cot(3*x)*sin(2*x)*tan(8*x)/(cos(6*x)*tan(4*x))
- cot(p*x)*log(x)
- Seno sin
- sin(2)^2/x
- sin(8*x)/(5*x)
- sin(12*x)/(3*x)
- sin(1/(-1+x))
- sin(x)^2+cos(x)
- Seno sin
- sin(2)^2/x
- sin(8*x)/(5*x)
- sin(12*x)/(3*x)
- sin(1/(-1+x))
- sin(x)^2+cos(x)
- Cotangente cot
- cot(2*x)*sin(5*x)
- cot(3*x)*sin(6*x)
- cot(x)*tan(x)
- cot(3*x)*sin(2*x)*tan(8*x)/(cos(6*x)*tan(4*x))
- cot(p*x)*log(x)
- Coseno cos
- cos(x)/(-1+x)
- cos(x)/cos(2*x)
- cos(pi*x)^(sin(pi*x)/x)
- cos(-1+e^(x^2))/(-1+cos(x))
- cos(2*pi/x)/(-1+3*x)
Límite de la función cot(x)^2*sin(x)^2*(-sin(x)^(-2*x)/cot(x)^2+cos(x))
Solución
Ha introducido
[src]
/ / -2*x \\
| 2 2 |-sin (x) ||
lim |cot (x)*sin (x)*|------------ + cos(x)||
x->0+| | 2 ||
\ \ cot (x) //
$$\lim_{x \to 0^+}\left(\sin^{2}{\left(x \right)} \cot^{2}{\left(x \right)} \left(\frac{\left(-1\right) \sin^{- 2 x}{\left(x \right)}}{\cot^{2}{\left(x \right)}} + \cos{\left(x \right)}\right)\right)$$
Limit((cot(x)^2*sin(x)^2)*((-sin(x)^(-2*x))/cot(x)^2 + cos(x)), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+} \sin^{2 - 2 x}{\left(x \right)} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\sin^{2 x}{\left(x \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} - 1} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\sin^{2}{\left(x \right)} \cot^{2}{\left(x \right)} \left(\frac{\left(-1\right) \sin^{- 2 x}{\left(x \right)}}{\cot^{2}{\left(x \right)}} + \cos{\left(x \right)}\right)\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\left(\sin^{2 x}{\left(x \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} - 1\right) \sin^{2}{\left(x \right)} \sin^{- 2 x}{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin^{2 - 2 x}{\left(x \right)}}{\frac{d}{d x} \frac{1}{\sin^{2 x}{\left(x \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} - 1}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(- 2 x \sin{\left(x \right)} \sin^{4 x}{\left(x \right)} \cos^{3}{\left(x \right)} \cot^{4}{\left(x \right)} + 4 x \sin{\left(x \right)} \sin^{2 x}{\left(x \right)} \cos^{2}{\left(x \right)} \cot^{2}{\left(x \right)} - 2 x \sin{\left(x \right)} \cos{\left(x \right)} - 2 \log{\left(\sin{\left(x \right)} \right)} \sin^{2}{\left(x \right)} \sin^{4 x}{\left(x \right)} \cos^{2}{\left(x \right)} \cot^{4}{\left(x \right)} + 4 \log{\left(\sin{\left(x \right)} \right)} \sin^{2}{\left(x \right)} \sin^{2 x}{\left(x \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} - 2 \log{\left(\sin{\left(x \right)} \right)} \sin^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \sin^{4 x}{\left(x \right)} \cos^{3}{\left(x \right)} \cot^{4}{\left(x \right)} - 4 \sin{\left(x \right)} \sin^{2 x}{\left(x \right)} \cos^{2}{\left(x \right)} \cot^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \sin^{- 2 x}{\left(x \right)}}{- \frac{2 x \sin^{2 x}{\left(x \right)} \cos^{2}{\left(x \right)} \cot^{2}{\left(x \right)}}{\sin{\left(x \right)}} - 2 \log{\left(\sin{\left(x \right)} \right)} \sin^{2 x}{\left(x \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} + \sin{\left(x \right)} \sin^{2 x}{\left(x \right)} \cot^{2}{\left(x \right)} + 2 \sin^{2 x}{\left(x \right)} \cos{\left(x \right)} \cot^{3}{\left(x \right)} + 2 \sin^{2 x}{\left(x \right)} \cos{\left(x \right)} \cot{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(- 2 x \sin{\left(x \right)} \sin^{4 x}{\left(x \right)} \cos^{3}{\left(x \right)} \cot^{4}{\left(x \right)} + 4 x \sin{\left(x \right)} \sin^{2 x}{\left(x \right)} \cos^{2}{\left(x \right)} \cot^{2}{\left(x \right)} - 2 x \sin{\left(x \right)} \cos{\left(x \right)} - 2 \log{\left(\sin{\left(x \right)} \right)} \sin^{2}{\left(x \right)} \sin^{4 x}{\left(x \right)} \cos^{2}{\left(x \right)} \cot^{4}{\left(x \right)} + 4 \log{\left(\sin{\left(x \right)} \right)} \sin^{2}{\left(x \right)} \sin^{2 x}{\left(x \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} - 2 \log{\left(\sin{\left(x \right)} \right)} \sin^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \sin^{4 x}{\left(x \right)} \cos^{3}{\left(x \right)} \cot^{4}{\left(x \right)} - 4 \sin{\left(x \right)} \sin^{2 x}{\left(x \right)} \cos^{2}{\left(x \right)} \cot^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \sin^{- 2 x}{\left(x \right)}}{- \frac{2 x \sin^{2 x}{\left(x \right)} \cos^{2}{\left(x \right)} \cot^{2}{\left(x \right)}}{\sin{\left(x \right)}} - 2 \log{\left(\sin{\left(x \right)} \right)} \sin^{2 x}{\left(x \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} + \sin{\left(x \right)} \sin^{2 x}{\left(x \right)} \cot^{2}{\left(x \right)} + 2 \sin^{2 x}{\left(x \right)} \cos{\left(x \right)} \cot^{3}{\left(x \right)} + 2 \sin^{2 x}{\left(x \right)} \cos{\left(x \right)} \cot{\left(x \right)}}\right)$$
=
$$1$$
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
$$\lim_{x \to 0^+} \sin^{2 - 2 x}{\left(x \right)} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\sin^{2 x}{\left(x \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} - 1} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\sin^{2}{\left(x \right)} \cot^{2}{\left(x \right)} \left(\frac{\left(-1\right) \sin^{- 2 x}{\left(x \right)}}{\cot^{2}{\left(x \right)}} + \cos{\left(x \right)}\right)\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\left(\sin^{2 x}{\left(x \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} - 1\right) \sin^{2}{\left(x \right)} \sin^{- 2 x}{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin^{2 - 2 x}{\left(x \right)}}{\frac{d}{d x} \frac{1}{\sin^{2 x}{\left(x \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} - 1}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(- 2 x \sin{\left(x \right)} \sin^{4 x}{\left(x \right)} \cos^{3}{\left(x \right)} \cot^{4}{\left(x \right)} + 4 x \sin{\left(x \right)} \sin^{2 x}{\left(x \right)} \cos^{2}{\left(x \right)} \cot^{2}{\left(x \right)} - 2 x \sin{\left(x \right)} \cos{\left(x \right)} - 2 \log{\left(\sin{\left(x \right)} \right)} \sin^{2}{\left(x \right)} \sin^{4 x}{\left(x \right)} \cos^{2}{\left(x \right)} \cot^{4}{\left(x \right)} + 4 \log{\left(\sin{\left(x \right)} \right)} \sin^{2}{\left(x \right)} \sin^{2 x}{\left(x \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} - 2 \log{\left(\sin{\left(x \right)} \right)} \sin^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \sin^{4 x}{\left(x \right)} \cos^{3}{\left(x \right)} \cot^{4}{\left(x \right)} - 4 \sin{\left(x \right)} \sin^{2 x}{\left(x \right)} \cos^{2}{\left(x \right)} \cot^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \sin^{- 2 x}{\left(x \right)}}{- \frac{2 x \sin^{2 x}{\left(x \right)} \cos^{2}{\left(x \right)} \cot^{2}{\left(x \right)}}{\sin{\left(x \right)}} - 2 \log{\left(\sin{\left(x \right)} \right)} \sin^{2 x}{\left(x \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} + \sin{\left(x \right)} \sin^{2 x}{\left(x \right)} \cot^{2}{\left(x \right)} + 2 \sin^{2 x}{\left(x \right)} \cos{\left(x \right)} \cot^{3}{\left(x \right)} + 2 \sin^{2 x}{\left(x \right)} \cos{\left(x \right)} \cot{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(- 2 x \sin{\left(x \right)} \sin^{4 x}{\left(x \right)} \cos^{3}{\left(x \right)} \cot^{4}{\left(x \right)} + 4 x \sin{\left(x \right)} \sin^{2 x}{\left(x \right)} \cos^{2}{\left(x \right)} \cot^{2}{\left(x \right)} - 2 x \sin{\left(x \right)} \cos{\left(x \right)} - 2 \log{\left(\sin{\left(x \right)} \right)} \sin^{2}{\left(x \right)} \sin^{4 x}{\left(x \right)} \cos^{2}{\left(x \right)} \cot^{4}{\left(x \right)} + 4 \log{\left(\sin{\left(x \right)} \right)} \sin^{2}{\left(x \right)} \sin^{2 x}{\left(x \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} - 2 \log{\left(\sin{\left(x \right)} \right)} \sin^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \sin^{4 x}{\left(x \right)} \cos^{3}{\left(x \right)} \cot^{4}{\left(x \right)} - 4 \sin{\left(x \right)} \sin^{2 x}{\left(x \right)} \cos^{2}{\left(x \right)} \cot^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \sin^{- 2 x}{\left(x \right)}}{- \frac{2 x \sin^{2 x}{\left(x \right)} \cos^{2}{\left(x \right)} \cot^{2}{\left(x \right)}}{\sin{\left(x \right)}} - 2 \log{\left(\sin{\left(x \right)} \right)} \sin^{2 x}{\left(x \right)} \cos{\left(x \right)} \cot^{2}{\left(x \right)} + \sin{\left(x \right)} \sin^{2 x}{\left(x \right)} \cot^{2}{\left(x \right)} + 2 \sin^{2 x}{\left(x \right)} \cos{\left(x \right)} \cot^{3}{\left(x \right)} + 2 \sin^{2 x}{\left(x \right)} \cos{\left(x \right)} \cot{\left(x \right)}}\right)$$
=
$$1$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 1 vez (veces)
Gráfica
A la izquierda y a la derecha
[src]
/ / -2*x \\
| 2 2 |-sin (x) ||
lim |cot (x)*sin (x)*|------------ + cos(x)||
x->0+| | 2 ||
\ \ cot (x) //
$$\lim_{x \to 0^+}\left(\sin^{2}{\left(x \right)} \cot^{2}{\left(x \right)} \left(\frac{\left(-1\right) \sin^{- 2 x}{\left(x \right)}}{\cot^{2}{\left(x \right)}} + \cos{\left(x \right)}\right)\right)$$
1
$$1$$
= 0.999999694003347
/ / -2*x \\
| 2 2 |-sin (x) ||
lim |cot (x)*sin (x)*|------------ + cos(x)||
x->0-| | 2 ||
\ \ cot (x) //
$$\lim_{x \to 0^-}\left(\sin^{2}{\left(x \right)} \cot^{2}{\left(x \right)} \left(\frac{\left(-1\right) \sin^{- 2 x}{\left(x \right)}}{\cot^{2}{\left(x \right)}} + \cos{\left(x \right)}\right)\right)$$
1
$$1$$
= (1.00000000007584 - 4.06043005141319e-13j)
= (1.00000000007584 - 4.06043005141319e-13j)
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\sin^{2}{\left(x \right)} \cot^{2}{\left(x \right)} \left(\frac{\left(-1\right) \sin^{- 2 x}{\left(x \right)}}{\cot^{2}{\left(x \right)}} + \cos{\left(x \right)}\right)\right) = 1$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\sin^{2}{\left(x \right)} \cot^{2}{\left(x \right)} \left(\frac{\left(-1\right) \sin^{- 2 x}{\left(x \right)}}{\cot^{2}{\left(x \right)}} + \cos{\left(x \right)}\right)\right) = 1$$
$$\lim_{x \to \infty}\left(\sin^{2}{\left(x \right)} \cot^{2}{\left(x \right)} \left(\frac{\left(-1\right) \sin^{- 2 x}{\left(x \right)}}{\cot^{2}{\left(x \right)}} + \cos{\left(x \right)}\right)\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\sin^{2}{\left(x \right)} \cot^{2}{\left(x \right)} \left(\frac{\left(-1\right) \sin^{- 2 x}{\left(x \right)}}{\cot^{2}{\left(x \right)}} + \cos{\left(x \right)}\right)\right) = \frac{- \tan^{2}{\left(1 \right)} + \sin^{2}{\left(1 \right)} \cos{\left(1 \right)}}{\tan^{2}{\left(1 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\sin^{2}{\left(x \right)} \cot^{2}{\left(x \right)} \left(\frac{\left(-1\right) \sin^{- 2 x}{\left(x \right)}}{\cot^{2}{\left(x \right)}} + \cos{\left(x \right)}\right)\right) = \frac{- \tan^{2}{\left(1 \right)} + \sin^{2}{\left(1 \right)} \cos{\left(1 \right)}}{\tan^{2}{\left(1 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\sin^{2}{\left(x \right)} \cot^{2}{\left(x \right)} \left(\frac{\left(-1\right) \sin^{- 2 x}{\left(x \right)}}{\cot^{2}{\left(x \right)}} + \cos{\left(x \right)}\right)\right)$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\sin^{2}{\left(x \right)} \cot^{2}{\left(x \right)} \left(\frac{\left(-1\right) \sin^{- 2 x}{\left(x \right)}}{\cot^{2}{\left(x \right)}} + \cos{\left(x \right)}\right)\right) = 1$$
$$\lim_{x \to \infty}\left(\sin^{2}{\left(x \right)} \cot^{2}{\left(x \right)} \left(\frac{\left(-1\right) \sin^{- 2 x}{\left(x \right)}}{\cot^{2}{\left(x \right)}} + \cos{\left(x \right)}\right)\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\sin^{2}{\left(x \right)} \cot^{2}{\left(x \right)} \left(\frac{\left(-1\right) \sin^{- 2 x}{\left(x \right)}}{\cot^{2}{\left(x \right)}} + \cos{\left(x \right)}\right)\right) = \frac{- \tan^{2}{\left(1 \right)} + \sin^{2}{\left(1 \right)} \cos{\left(1 \right)}}{\tan^{2}{\left(1 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\sin^{2}{\left(x \right)} \cot^{2}{\left(x \right)} \left(\frac{\left(-1\right) \sin^{- 2 x}{\left(x \right)}}{\cot^{2}{\left(x \right)}} + \cos{\left(x \right)}\right)\right) = \frac{- \tan^{2}{\left(1 \right)} + \sin^{2}{\left(1 \right)} \cos{\left(1 \right)}}{\tan^{2}{\left(1 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\sin^{2}{\left(x \right)} \cot^{2}{\left(x \right)} \left(\frac{\left(-1\right) \sin^{- 2 x}{\left(x \right)}}{\cot^{2}{\left(x \right)}} + \cos{\left(x \right)}\right)\right)$$
Más detalles con x→-oo