Sr Examen
Otras calculadoras:
-
¿Cómo usar?
- Límite de la función:
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Límite de (-cos(2*x)+cos(x))/(1-cos(x))
-
Límite de x^2*(-cos(4*x)/3+cos(2*x)/3)
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Límite de ((-1+5*x^2)/(-1+7*x^2))^(1+5*x)
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Límite de (-1-5*x+4*x^2)/(-1-x+2*x^2)
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Expresiones idénticas
- (tres *sin(x)- tres *sin(dos *x)/ dos -x*(-cos(dos *x)+cos(x)))/(uno -cos(x))^ dos
- (3 multiplicar por seno de (x) menos 3 multiplicar por seno de (2 multiplicar por x) dividir por 2 menos x multiplicar por ( menos coseno de (2 multiplicar por x) más coseno de (x))) dividir por (1 menos coseno de (x)) al cuadrado
- (tres multiplicar por seno de (x) menos tres multiplicar por seno de (dos multiplicar por x) dividir por dos menos x multiplicar por ( menos coseno de (dos multiplicar por x) más coseno de (x))) dividir por (uno menos coseno de (x)) en el grado dos
- (3*sin(x)-3*sin(2*x)/2-x*(-cos(2*x)+cos(x)))/(1-cos(x))2
- 3*sinx-3*sin2*x/2-x*-cos2*x+cosx/1-cosx2
- (3*sin(x)-3*sin(2*x)/2-x*(-cos(2*x)+cos(x)))/(1-cos(x))²
- (3*sin(x)-3*sin(2*x)/2-x*(-cos(2*x)+cos(x)))/(1-cos(x)) en el grado 2
- (3sin(x)-3sin(2x)/2-x(-cos(2x)+cos(x)))/(1-cos(x))^2
- (3sin(x)-3sin(2x)/2-x(-cos(2x)+cos(x)))/(1-cos(x))2
- 3sinx-3sin2x/2-x-cos2x+cosx/1-cosx2
- 3sinx-3sin2x/2-x-cos2x+cosx/1-cosx^2
- (3*sin(x)-3*sin(2*x) dividir por 2-x*(-cos(2*x)+cos(x))) dividir por (1-cos(x))^2
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Expresiones semejantes
- (3*sin(x)-3*sin(2*x)/2+x*(-cos(2*x)+cos(x)))/(1-cos(x))^2
- (3*sin(x)+3*sin(2*x)/2-x*(-cos(2*x)+cos(x)))/(1-cos(x))^2
- (3*sin(x)-3*sin(2*x)/2-x*(cos(2*x)+cos(x)))/(1-cos(x))^2
- (3*sin(x)-3*sin(2*x)/2-x*(-cos(2*x)+cos(x)))/(1+cos(x))^2
- (3*sin(x)-3*sin(2*x)/2-x*(-cos(2*x)-cos(x)))/(1-cos(x))^2
- (3*sinx-3*sin(2*x)/2-x*(-cos(2*x)+cosx))/(1-cosx)^2
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Expresiones con funciones
- Seno sin
- sin(x)/cos(3*x)+cos(x)+sin(3*x)
- sin(x)^56
- sin(-sin(3*x)+5*x)/x
- sin(3*x)^2/atan(x)^23
- sin(7)^2*cos(x)
- Seno sin
- sin(x)/cos(3*x)+cos(x)+sin(3*x)
- sin(x)^56
- sin(-sin(3*x)+5*x)/x
- sin(3*x)^2/atan(x)^23
- sin(7)^2*cos(x)
- Coseno cos
- cos(2*x+pi/3)/sin(3*x)
- cos(x)^3+sin(x)^2
- cos(3*x)^(x^2)
- cos(3/x)^9
- cos(pi*x/(3+x^2))
- Coseno cos
- cos(2*x+pi/3)/sin(3*x)
- cos(x)^3+sin(x)^2
- cos(3*x)^(x^2)
- cos(3/x)^9
- cos(pi*x/(3+x^2))
- Coseno cos
- cos(2*x+pi/3)/sin(3*x)
- cos(x)^3+sin(x)^2
- cos(3*x)^(x^2)
- cos(3/x)^9
- cos(pi*x/(3+x^2))
Límite de la función (3*sin(x)-3*sin(2*x)/2-x*(-cos(2*x)+cos(x)))/(1-cos(x))^2
Solución
Ha introducido
[src]
/ 3*sin(2*x) \
|3*sin(x) - ---------- - x*(-cos(2*x) + cos(x))|
| 2 |
lim |----------------------------------------------|
x->0+| 2 |
\ (1 - cos(x)) /
$$\lim_{x \to 0^+}\left(\frac{- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(3 \sin{\left(x \right)} - \frac{3 \sin{\left(2 x \right)}}{2}\right)}{\left(1 - \cos{\left(x \right)}\right)^{2}}\right)$$
Limit((3*sin(x) - 3*sin(2*x)/2 - x*(-cos(2*x) + cos(x)))/(1 - cos(x))^2, x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(- 2 x \cos{\left(x \right)} + 2 x \cos{\left(2 x \right)} + 6 \sin{\left(x \right)} - 3 \sin{\left(2 x \right)}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+}\left(2 \cos^{2}{\left(x \right)} - 4 \cos{\left(x \right)} + 2\right) = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(3 \sin{\left(x \right)} - \frac{3 \sin{\left(2 x \right)}}{2}\right)}{\left(1 - \cos{\left(x \right)}\right)^{2}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\frac{- 2 x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + 6 \sin{\left(x \right)} - 3 \sin{\left(2 x \right)}}{2 \left(1 - \cos{\left(x \right)}\right)^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- 2 x \cos{\left(x \right)} + 2 x \cos{\left(2 x \right)} + 6 \sin{\left(x \right)} - 3 \sin{\left(2 x \right)}\right)}{\frac{d}{d x} \left(2 \cos^{2}{\left(x \right)} - 4 \cos{\left(x \right)} + 2\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 x \sin{\left(x \right)} - 4 x \sin{\left(2 x \right)} + 4 \cos{\left(x \right)} - 4 \cos{\left(2 x \right)}}{- 4 \sin{\left(x \right)} \cos{\left(x \right)} + 4 \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(2 x \sin{\left(x \right)} - 4 x \sin{\left(2 x \right)} + 4 \cos{\left(x \right)} - 4 \cos{\left(2 x \right)}\right)}{\frac{d}{d x} \left(- 4 \sin{\left(x \right)} \cos{\left(x \right)} + 4 \sin{\left(x \right)}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 x \cos{\left(x \right)} - 8 x \cos{\left(2 x \right)} - 2 \sin{\left(x \right)} + 4 \sin{\left(2 x \right)}}{4 \sin^{2}{\left(x \right)} - 4 \cos^{2}{\left(x \right)} + 4 \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(2 x \cos{\left(x \right)} - 8 x \cos{\left(2 x \right)} - 2 \sin{\left(x \right)} + 4 \sin{\left(2 x \right)}\right)}{\frac{d}{d x} \left(4 \sin^{2}{\left(x \right)} - 4 \cos^{2}{\left(x \right)} + 4 \cos{\left(x \right)}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- 2 x \sin{\left(x \right)} + 16 x \sin{\left(2 x \right)}}{16 \sin{\left(x \right)} \cos{\left(x \right)} - 4 \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- 2 x \sin{\left(x \right)} + 16 x \sin{\left(2 x \right)}}{16 \sin{\left(x \right)} \cos{\left(x \right)} - 4 \sin{\left(x \right)}}\right)$$
=
$$0$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)
0/0,
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(- 2 x \cos{\left(x \right)} + 2 x \cos{\left(2 x \right)} + 6 \sin{\left(x \right)} - 3 \sin{\left(2 x \right)}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+}\left(2 \cos^{2}{\left(x \right)} - 4 \cos{\left(x \right)} + 2\right) = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(3 \sin{\left(x \right)} - \frac{3 \sin{\left(2 x \right)}}{2}\right)}{\left(1 - \cos{\left(x \right)}\right)^{2}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\frac{- 2 x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + 6 \sin{\left(x \right)} - 3 \sin{\left(2 x \right)}}{2 \left(1 - \cos{\left(x \right)}\right)^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- 2 x \cos{\left(x \right)} + 2 x \cos{\left(2 x \right)} + 6 \sin{\left(x \right)} - 3 \sin{\left(2 x \right)}\right)}{\frac{d}{d x} \left(2 \cos^{2}{\left(x \right)} - 4 \cos{\left(x \right)} + 2\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 x \sin{\left(x \right)} - 4 x \sin{\left(2 x \right)} + 4 \cos{\left(x \right)} - 4 \cos{\left(2 x \right)}}{- 4 \sin{\left(x \right)} \cos{\left(x \right)} + 4 \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(2 x \sin{\left(x \right)} - 4 x \sin{\left(2 x \right)} + 4 \cos{\left(x \right)} - 4 \cos{\left(2 x \right)}\right)}{\frac{d}{d x} \left(- 4 \sin{\left(x \right)} \cos{\left(x \right)} + 4 \sin{\left(x \right)}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 x \cos{\left(x \right)} - 8 x \cos{\left(2 x \right)} - 2 \sin{\left(x \right)} + 4 \sin{\left(2 x \right)}}{4 \sin^{2}{\left(x \right)} - 4 \cos^{2}{\left(x \right)} + 4 \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(2 x \cos{\left(x \right)} - 8 x \cos{\left(2 x \right)} - 2 \sin{\left(x \right)} + 4 \sin{\left(2 x \right)}\right)}{\frac{d}{d x} \left(4 \sin^{2}{\left(x \right)} - 4 \cos^{2}{\left(x \right)} + 4 \cos{\left(x \right)}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- 2 x \sin{\left(x \right)} + 16 x \sin{\left(2 x \right)}}{16 \sin{\left(x \right)} \cos{\left(x \right)} - 4 \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- 2 x \sin{\left(x \right)} + 16 x \sin{\left(2 x \right)}}{16 \sin{\left(x \right)} \cos{\left(x \right)} - 4 \sin{\left(x \right)}}\right)$$
=
$$0$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)
Gráfica
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(3 \sin{\left(x \right)} - \frac{3 \sin{\left(2 x \right)}}{2}\right)}{\left(1 - \cos{\left(x \right)}\right)^{2}}\right) = 0$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(3 \sin{\left(x \right)} - \frac{3 \sin{\left(2 x \right)}}{2}\right)}{\left(1 - \cos{\left(x \right)}\right)^{2}}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(3 \sin{\left(x \right)} - \frac{3 \sin{\left(2 x \right)}}{2}\right)}{\left(1 - \cos{\left(x \right)}\right)^{2}}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(3 \sin{\left(x \right)} - \frac{3 \sin{\left(2 x \right)}}{2}\right)}{\left(1 - \cos{\left(x \right)}\right)^{2}}\right) = - \frac{- 6 \sin{\left(1 \right)} - 2 \cos{\left(2 \right)} + 2 \cos{\left(1 \right)} + 3 \sin{\left(2 \right)}}{- 4 \cos{\left(1 \right)} + 2 \cos^{2}{\left(1 \right)} + 2}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(3 \sin{\left(x \right)} - \frac{3 \sin{\left(2 x \right)}}{2}\right)}{\left(1 - \cos{\left(x \right)}\right)^{2}}\right) = - \frac{- 6 \sin{\left(1 \right)} - 2 \cos{\left(2 \right)} + 2 \cos{\left(1 \right)} + 3 \sin{\left(2 \right)}}{- 4 \cos{\left(1 \right)} + 2 \cos^{2}{\left(1 \right)} + 2}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(3 \sin{\left(x \right)} - \frac{3 \sin{\left(2 x \right)}}{2}\right)}{\left(1 - \cos{\left(x \right)}\right)^{2}}\right)$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(3 \sin{\left(x \right)} - \frac{3 \sin{\left(2 x \right)}}{2}\right)}{\left(1 - \cos{\left(x \right)}\right)^{2}}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(3 \sin{\left(x \right)} - \frac{3 \sin{\left(2 x \right)}}{2}\right)}{\left(1 - \cos{\left(x \right)}\right)^{2}}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(3 \sin{\left(x \right)} - \frac{3 \sin{\left(2 x \right)}}{2}\right)}{\left(1 - \cos{\left(x \right)}\right)^{2}}\right) = - \frac{- 6 \sin{\left(1 \right)} - 2 \cos{\left(2 \right)} + 2 \cos{\left(1 \right)} + 3 \sin{\left(2 \right)}}{- 4 \cos{\left(1 \right)} + 2 \cos^{2}{\left(1 \right)} + 2}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(3 \sin{\left(x \right)} - \frac{3 \sin{\left(2 x \right)}}{2}\right)}{\left(1 - \cos{\left(x \right)}\right)^{2}}\right) = - \frac{- 6 \sin{\left(1 \right)} - 2 \cos{\left(2 \right)} + 2 \cos{\left(1 \right)} + 3 \sin{\left(2 \right)}}{- 4 \cos{\left(1 \right)} + 2 \cos^{2}{\left(1 \right)} + 2}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(3 \sin{\left(x \right)} - \frac{3 \sin{\left(2 x \right)}}{2}\right)}{\left(1 - \cos{\left(x \right)}\right)^{2}}\right)$$
Más detalles con x→-oo
A la izquierda y a la derecha
[src]
/ 3*sin(2*x) \
|3*sin(x) - ---------- - x*(-cos(2*x) + cos(x))|
| 2 |
lim |----------------------------------------------|
x->0+| 2 |
\ (1 - cos(x)) /
$$\lim_{x \to 0^+}\left(\frac{- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(3 \sin{\left(x \right)} - \frac{3 \sin{\left(2 x \right)}}{2}\right)}{\left(1 - \cos{\left(x \right)}\right)^{2}}\right)$$
0
$$0$$
= -1.44937556952754e-32
/ 3*sin(2*x) \
|3*sin(x) - ---------- - x*(-cos(2*x) + cos(x))|
| 2 |
lim |----------------------------------------------|
x->0-| 2 |
\ (1 - cos(x)) /
$$\lim_{x \to 0^-}\left(\frac{- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(3 \sin{\left(x \right)} - \frac{3 \sin{\left(2 x \right)}}{2}\right)}{\left(1 - \cos{\left(x \right)}\right)^{2}}\right)$$
0
$$0$$
= 1.44937556952754e-32
= 1.44937556952754e-32