Sr Examen
Otras calculadoras:
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- Límite de la función:
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Límite de (5-x^2+2*x^3+3*x^4+5*x)/(1+x^3)
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Límite de ((27+x)^(1/3)-(27-x)^(1/3))/(x+2*(x^4)^(1/3))
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Límite de ((1-x)^4-(1+x)^4)/((1+x)^3-(1-x)^3)
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Límite de ((1+x)/(2+x))^((1-sqrt(x))/(1-x))
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Expresiones idénticas
- x^ dos *(-sin(dos *x)+ dos *sin(x)-x*(-cos(dos *x)+cos(x)))/((uno -cos(x))^ dos *sin(x)^ dos)
- x al cuadrado multiplicar por ( menos seno de (2 multiplicar por x) más 2 multiplicar por seno de (x) menos x multiplicar por ( menos coseno de (2 multiplicar por x) más coseno de (x))) dividir por ((1 menos coseno de (x)) al cuadrado multiplicar por seno de (x) al cuadrado )
- x en el grado dos multiplicar por ( menos seno de (dos multiplicar por x) más dos multiplicar por seno de (x) 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 multiplicar por seno de (x) en el grado dos)
- x2*(-sin(2*x)+2*sin(x)-x*(-cos(2*x)+cos(x)))/((1-cos(x))2*sin(x)2)
- x2*-sin2*x+2*sinx-x*-cos2*x+cosx/1-cosx2*sinx2
- x²*(-sin(2*x)+2*sin(x)-x*(-cos(2*x)+cos(x)))/((1-cos(x))²*sin(x)²)
- x en el grado 2*(-sin(2*x)+2*sin(x)-x*(-cos(2*x)+cos(x)))/((1-cos(x)) en el grado 2*sin(x) en el grado 2)
- x^2(-sin(2x)+2sin(x)-x(-cos(2x)+cos(x)))/((1-cos(x))^2sin(x)^2)
- x2(-sin(2x)+2sin(x)-x(-cos(2x)+cos(x)))/((1-cos(x))2sin(x)2)
- x2-sin2x+2sinx-x-cos2x+cosx/1-cosx2sinx2
- x^2-sin2x+2sinx-x-cos2x+cosx/1-cosx^2sinx^2
- x^2*(-sin(2*x)+2*sin(x)-x*(-cos(2*x)+cos(x))) dividir por ((1-cos(x))^2*sin(x)^2)
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Expresiones semejantes
- x^2*(-sin(2*x)+2*sin(x)+x*(-cos(2*x)+cos(x)))/((1-cos(x))^2*sin(x)^2)
- x^2*(-sin(2*x)+2*sin(x)-x*(cos(2*x)+cos(x)))/((1-cos(x))^2*sin(x)^2)
- x^2*(-sin(2*x)-2*sin(x)-x*(-cos(2*x)+cos(x)))/((1-cos(x))^2*sin(x)^2)
- x^2*(sin(2*x)+2*sin(x)-x*(-cos(2*x)+cos(x)))/((1-cos(x))^2*sin(x)^2)
- x^2*(-sin(2*x)+2*sin(x)-x*(-cos(2*x)+cos(x)))/((1+cos(x))^2*sin(x)^2)
- x^2*(-sin(2*x)+2*sin(x)-x*(-cos(2*x)-cos(x)))/((1-cos(x))^2*sin(x)^2)
- x^2*(-sin(2*x)+2*sinx-x*(-cos(2*x)+cosx))/((1-cosx)^2*sinx^2)
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Expresiones con funciones
- Seno sin
- sin(5/x)/(3+x^6)
- sin(x)*tan(x/4)/((-1+e^(2*x))*log(1-2*x))
- sin(31*x)/(30*x)
- sin(5*x)/tan(3*z)
- sin(n*p)/n
- Seno sin
- sin(5/x)/(3+x^6)
- sin(x)*tan(x/4)/((-1+e^(2*x))*log(1-2*x))
- sin(31*x)/(30*x)
- sin(5*x)/tan(3*z)
- sin(n*p)/n
- Coseno cos
- cos(x)/x2
- cos(x+5*p/2)*tan(x)/asin(2*x^2)
- cos(x)*log(x)/log(-1+e^x)
- cos(4*x)*sin(8*x)/(x*sin(4*x))
- cos(x*2^n)
- Coseno cos
- cos(x)/x2
- cos(x+5*p/2)*tan(x)/asin(2*x^2)
- cos(x)*log(x)/log(-1+e^x)
- cos(4*x)*sin(8*x)/(x*sin(4*x))
- cos(x*2^n)
- Coseno cos
- cos(x)/x2
- cos(x+5*p/2)*tan(x)/asin(2*x^2)
- cos(x)*log(x)/log(-1+e^x)
- cos(4*x)*sin(8*x)/(x*sin(4*x))
- cos(x*2^n)
- Seno sin
- sin(5/x)/(3+x^6)
- sin(x)*tan(x/4)/((-1+e^(2*x))*log(1-2*x))
- sin(31*x)/(30*x)
- sin(5*x)/tan(3*z)
- sin(n*p)/n
Límite de la función x^2*(-sin(2*x)+2*sin(x)-x*(-cos(2*x)+cos(x)))/((1-cos(x))^2*sin(x)^2)
Solución
Ha introducido
[src]
/ 2 \
|x *(-sin(2*x) + 2*sin(x) - x*(-cos(2*x) + cos(x)))|
lim |--------------------------------------------------|
x->0+| 2 2 |
\ (1 - cos(x)) *sin (x) /
$$\lim_{x \to 0^+}\left(\frac{x^{2} \left(- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(2 \sin{\left(x \right)} - \sin{\left(2 x \right)}\right)\right)}{\left(1 - \cos{\left(x \right)}\right)^{2} \sin^{2}{\left(x \right)}}\right)$$
Limit((x^2*(-sin(2*x) + 2*sin(x) - x*(-cos(2*x) + cos(x))))/(((1 - cos(x))^2*sin(x)^2)), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(- x \cos{\left(x \right)} + x \cos{\left(2 x \right)} + 2 \sin{\left(x \right)} - \sin{\left(2 x \right)}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+}\left(\frac{\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{x^{2}} - \frac{2 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{x^{2}} + \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right) = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{x^{2} \left(- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(2 \sin{\left(x \right)} - \sin{\left(2 x \right)}\right)\right)}{\left(1 - \cos{\left(x \right)}\right)^{2} \sin^{2}{\left(x \right)}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\frac{x^{2} \left(- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + 2 \sin{\left(x \right)} - \sin{\left(2 x \right)}\right)}{\left(1 - \cos{\left(x \right)}\right)^{2} \sin^{2}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- x \cos{\left(x \right)} + x \cos{\left(2 x \right)} + 2 \sin{\left(x \right)} - \sin{\left(2 x \right)}\right)}{\frac{d}{d x} \left(\frac{\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{x^{2}} - \frac{2 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{x^{2}} + \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{x \sin{\left(x \right)} - 2 x \sin{\left(2 x \right)} + \cos{\left(x \right)} - \cos{\left(2 x \right)}}{- \frac{2 \sin^{3}{\left(x \right)} \cos{\left(x \right)}}{x^{2}} + \frac{2 \sin^{3}{\left(x \right)}}{x^{2}} + \frac{2 \sin{\left(x \right)} \cos^{3}{\left(x \right)}}{x^{2}} - \frac{4 \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{x^{2}} + \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{x^{2}} - \frac{2 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{x^{3}} + \frac{4 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{x^{3}} - \frac{2 \sin^{2}{\left(x \right)}}{x^{3}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{x \sin{\left(x \right)} - 2 x \sin{\left(2 x \right)} + \cos{\left(x \right)} - \cos{\left(2 x \right)}}{- \frac{2 \sin^{3}{\left(x \right)} \cos{\left(x \right)}}{x^{2}} + \frac{2 \sin^{3}{\left(x \right)}}{x^{2}} + \frac{2 \sin{\left(x \right)} \cos^{3}{\left(x \right)}}{x^{2}} - \frac{4 \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{x^{2}} + \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{x^{2}} - \frac{2 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{x^{3}} + \frac{4 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{x^{3}} - \frac{2 \sin^{2}{\left(x \right)}}{x^{3}}}\right)$$
=
$$-\infty$$
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
$$\lim_{x \to 0^+}\left(- x \cos{\left(x \right)} + x \cos{\left(2 x \right)} + 2 \sin{\left(x \right)} - \sin{\left(2 x \right)}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+}\left(\frac{\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{x^{2}} - \frac{2 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{x^{2}} + \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right) = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{x^{2} \left(- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(2 \sin{\left(x \right)} - \sin{\left(2 x \right)}\right)\right)}{\left(1 - \cos{\left(x \right)}\right)^{2} \sin^{2}{\left(x \right)}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\frac{x^{2} \left(- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + 2 \sin{\left(x \right)} - \sin{\left(2 x \right)}\right)}{\left(1 - \cos{\left(x \right)}\right)^{2} \sin^{2}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- x \cos{\left(x \right)} + x \cos{\left(2 x \right)} + 2 \sin{\left(x \right)} - \sin{\left(2 x \right)}\right)}{\frac{d}{d x} \left(\frac{\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{x^{2}} - \frac{2 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{x^{2}} + \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{x \sin{\left(x \right)} - 2 x \sin{\left(2 x \right)} + \cos{\left(x \right)} - \cos{\left(2 x \right)}}{- \frac{2 \sin^{3}{\left(x \right)} \cos{\left(x \right)}}{x^{2}} + \frac{2 \sin^{3}{\left(x \right)}}{x^{2}} + \frac{2 \sin{\left(x \right)} \cos^{3}{\left(x \right)}}{x^{2}} - \frac{4 \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{x^{2}} + \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{x^{2}} - \frac{2 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{x^{3}} + \frac{4 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{x^{3}} - \frac{2 \sin^{2}{\left(x \right)}}{x^{3}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{x \sin{\left(x \right)} - 2 x \sin{\left(2 x \right)} + \cos{\left(x \right)} - \cos{\left(2 x \right)}}{- \frac{2 \sin^{3}{\left(x \right)} \cos{\left(x \right)}}{x^{2}} + \frac{2 \sin^{3}{\left(x \right)}}{x^{2}} + \frac{2 \sin{\left(x \right)} \cos^{3}{\left(x \right)}}{x^{2}} - \frac{4 \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{x^{2}} + \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{x^{2}} - \frac{2 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{x^{3}} + \frac{4 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{x^{3}} - \frac{2 \sin^{2}{\left(x \right)}}{x^{3}}}\right)$$
=
$$-\infty$$
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 *(-sin(2*x) + 2*sin(x) - x*(-cos(2*x) + cos(x)))|
lim |--------------------------------------------------|
x->0+| 2 2 |
\ (1 - cos(x)) *sin (x) /
$$\lim_{x \to 0^+}\left(\frac{x^{2} \left(- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(2 \sin{\left(x \right)} - \sin{\left(2 x \right)}\right)\right)}{\left(1 - \cos{\left(x \right)}\right)^{2} \sin^{2}{\left(x \right)}}\right)$$
-oo
$$-\infty$$
= -301.99668867637
/ 2 \
|x *(-sin(2*x) + 2*sin(x) - x*(-cos(2*x) + cos(x)))|
lim |--------------------------------------------------|
x->0-| 2 2 |
\ (1 - cos(x)) *sin (x) /
$$\lim_{x \to 0^-}\left(\frac{x^{2} \left(- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(2 \sin{\left(x \right)} - \sin{\left(2 x \right)}\right)\right)}{\left(1 - \cos{\left(x \right)}\right)^{2} \sin^{2}{\left(x \right)}}\right)$$
oo
$$\infty$$
= 301.99668867637
= 301.99668867637
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{x^{2} \left(- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(2 \sin{\left(x \right)} - \sin{\left(2 x \right)}\right)\right)}{\left(1 - \cos{\left(x \right)}\right)^{2} \sin^{2}{\left(x \right)}}\right) = -\infty$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{x^{2} \left(- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(2 \sin{\left(x \right)} - \sin{\left(2 x \right)}\right)\right)}{\left(1 - \cos{\left(x \right)}\right)^{2} \sin^{2}{\left(x \right)}}\right) = -\infty$$
$$\lim_{x \to \infty}\left(\frac{x^{2} \left(- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(2 \sin{\left(x \right)} - \sin{\left(2 x \right)}\right)\right)}{\left(1 - \cos{\left(x \right)}\right)^{2} \sin^{2}{\left(x \right)}}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{x^{2} \left(- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(2 \sin{\left(x \right)} - \sin{\left(2 x \right)}\right)\right)}{\left(1 - \cos{\left(x \right)}\right)^{2} \sin^{2}{\left(x \right)}}\right) = - \frac{- 2 \sin{\left(1 \right)} - \cos{\left(2 \right)} + \cos{\left(1 \right)} + \sin{\left(2 \right)}}{- 2 \sin^{2}{\left(1 \right)} \cos{\left(1 \right)} + \sin^{2}{\left(1 \right)} \cos^{2}{\left(1 \right)} + \sin^{2}{\left(1 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{x^{2} \left(- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(2 \sin{\left(x \right)} - \sin{\left(2 x \right)}\right)\right)}{\left(1 - \cos{\left(x \right)}\right)^{2} \sin^{2}{\left(x \right)}}\right) = - \frac{- 2 \sin{\left(1 \right)} - \cos{\left(2 \right)} + \cos{\left(1 \right)} + \sin{\left(2 \right)}}{- 2 \sin^{2}{\left(1 \right)} \cos{\left(1 \right)} + \sin^{2}{\left(1 \right)} \cos^{2}{\left(1 \right)} + \sin^{2}{\left(1 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{x^{2} \left(- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(2 \sin{\left(x \right)} - \sin{\left(2 x \right)}\right)\right)}{\left(1 - \cos{\left(x \right)}\right)^{2} \sin^{2}{\left(x \right)}}\right)$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{x^{2} \left(- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(2 \sin{\left(x \right)} - \sin{\left(2 x \right)}\right)\right)}{\left(1 - \cos{\left(x \right)}\right)^{2} \sin^{2}{\left(x \right)}}\right) = -\infty$$
$$\lim_{x \to \infty}\left(\frac{x^{2} \left(- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(2 \sin{\left(x \right)} - \sin{\left(2 x \right)}\right)\right)}{\left(1 - \cos{\left(x \right)}\right)^{2} \sin^{2}{\left(x \right)}}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{x^{2} \left(- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(2 \sin{\left(x \right)} - \sin{\left(2 x \right)}\right)\right)}{\left(1 - \cos{\left(x \right)}\right)^{2} \sin^{2}{\left(x \right)}}\right) = - \frac{- 2 \sin{\left(1 \right)} - \cos{\left(2 \right)} + \cos{\left(1 \right)} + \sin{\left(2 \right)}}{- 2 \sin^{2}{\left(1 \right)} \cos{\left(1 \right)} + \sin^{2}{\left(1 \right)} \cos^{2}{\left(1 \right)} + \sin^{2}{\left(1 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{x^{2} \left(- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(2 \sin{\left(x \right)} - \sin{\left(2 x \right)}\right)\right)}{\left(1 - \cos{\left(x \right)}\right)^{2} \sin^{2}{\left(x \right)}}\right) = - \frac{- 2 \sin{\left(1 \right)} - \cos{\left(2 \right)} + \cos{\left(1 \right)} + \sin{\left(2 \right)}}{- 2 \sin^{2}{\left(1 \right)} \cos{\left(1 \right)} + \sin^{2}{\left(1 \right)} \cos^{2}{\left(1 \right)} + \sin^{2}{\left(1 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{x^{2} \left(- x \left(\cos{\left(x \right)} - \cos{\left(2 x \right)}\right) + \left(2 \sin{\left(x \right)} - \sin{\left(2 x \right)}\right)\right)}{\left(1 - \cos{\left(x \right)}\right)^{2} \sin^{2}{\left(x \right)}}\right)$$
Más detalles con x→-oo