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Límite de (-3+x^2-2*x)/(-15-4*x+3*x^2)
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Límite de (sqrt(5+x)-sqrt(10))/(-15+x^2-2*x)
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Límite de (sqrt(-1+x)-sqrt(7-x))/(-4+x)
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Límite de (-2+sqrt(-2+3*x))/(sqrt(x)-sqrt(2))
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Expresiones idénticas
- tan(doce *x)^ ocho *asin(siete *x^ dos)/(atan(x^ ocho)*sin(ocho *x)^ dos)
- tangente de (12 multiplicar por x) en el grado 8 multiplicar por ar coseno de eno de (7 multiplicar por x al cuadrado ) dividir por ( arco tangente de gente de (x en el grado 8) multiplicar por seno de (8 multiplicar por x) al cuadrado )
- tangente de (doce multiplicar por x) en el grado ocho multiplicar por ar coseno de eno de (siete multiplicar por x en el grado dos) dividir por ( arco tangente de gente de (x en el grado ocho) multiplicar por seno de (ocho multiplicar por x) en el grado dos)
- tan(12*x)8*asin(7*x2)/(atan(x8)*sin(8*x)2)
- tan12*x8*asin7*x2/atanx8*sin8*x2
- tan(12*x)⁸*asin(7*x²)/(atan(x⁸)*sin(8*x)²)
- tan(12*x) en el grado 8*asin(7*x en el grado 2)/(atan(x en el grado 8)*sin(8*x) en el grado 2)
- tan(12x)^8asin(7x^2)/(atan(x^8)sin(8x)^2)
- tan(12x)8asin(7x2)/(atan(x8)sin(8x)2)
- tan12x8asin7x2/atanx8sin8x2
- tan12x^8asin7x^2/atanx^8sin8x^2
- tan(12*x)^8*asin(7*x^2) dividir por (atan(x^8)*sin(8*x)^2)
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Expresiones semejantes
- tan(12*x)^8*arcsin(7*x^2)/(arctan(x^8)*sin(8*x)^2)
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Expresiones con funciones
- Tangente tan
- tan(2*x)^2/(1-cos(x))
- tan(157*x/50)/x
- tan(-1+x^2)^3*log(1+x)/((-1+e^(-1+x))*(-1+sqrt(x))^2)
- tan(x)/(4*z^2-pi*x)
- tan(x)^(-cos(x))
- Arcoseno arcsin
- asin(2*x)^2
- asin(x)/|x|
- asin(3*x)*sin(x)/(x*acos(x)*tan(x))
- asin(-36+x^2)/(6+x)
- asin(-4+4*x^2)^2/sin(-1+x)^2
- Arcotangente arctan
- atan(-3+(27-x-2*x^2)^(1/3))/x
- atan(4*x)/x+(-1+x)/(-125+x^3)
- atan(sqrt(x))/sqrt(x)
- atan(sqrt((1+x)/(1+2*x)))
- atan(5*x)^2*(-1+e^(2*x^3))/log(1+3*x^5)
- Seno sin
- sin(15*x)/(5*x)
- sin(-8/7+x/7)*tan(pi*x/16)
- sin(x)^2/x^6
- sin(x)^26*tan(x)/(2*x)
- sin(pi*x)/(-log(pi)+log(pi*x))
Límite de la función tan(12*x)^8*asin(7*x^2)/(atan(x^8)*sin(8*x)^2)
Solución
Ha introducido
[src]
/ 8 / 2\\
|tan (12*x)*asin\7*x /|
lim |---------------------|
x->0+| / 8\ 2 |
\ atan\x /*sin (8*x) /
$$\lim_{x \to 0^+}\left(\frac{\tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)} \operatorname{atan}{\left(x^{8} \right)}}\right)$$
Limit((tan(12*x)^8*asin(7*x^2))/((atan(x^8)*sin(8*x)^2)), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(\frac{\tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)}}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \operatorname{atan}{\left(x^{8} \right)} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{\tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)} \operatorname{atan}{\left(x^{8} \right)}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\frac{\tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)} \operatorname{atan}{\left(x^{8} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{\tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)}}}{\frac{d}{d x} \operatorname{atan}{\left(x^{8} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(x^{16} + 1\right) \left(\frac{14 x \tan^{8}{\left(12 x \right)}}{\sqrt{1 - 49 x^{4}} \sin^{2}{\left(8 x \right)}} + \frac{\left(96 \tan^{2}{\left(12 x \right)} + 96\right) \tan^{7}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)}} - \frac{16 \cos{\left(8 x \right)} \tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{3}{\left(8 x \right)}}\right)}{8 x^{7}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{14 x \tan^{8}{\left(12 x \right)}}{\sqrt{1 - 49 x^{4}} \sin^{2}{\left(8 x \right)}} + \frac{96 \tan^{9}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)}} + \frac{96 \tan^{7}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)}} - \frac{16 \cos{\left(8 x \right)} \tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{3}{\left(8 x \right)}}}{8 x^{7}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{14 x \tan^{8}{\left(12 x \right)}}{\sqrt{1 - 49 x^{4}} \sin^{2}{\left(8 x \right)}} + \frac{96 \tan^{9}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)}} + \frac{96 \tan^{7}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)}} - \frac{16 \cos{\left(8 x \right)} \tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{3}{\left(8 x \right)}}}{8 x^{7}}\right)$$
=
$$47029248$$
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
$$\lim_{x \to 0^+}\left(\frac{\tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)}}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \operatorname{atan}{\left(x^{8} \right)} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{\tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)} \operatorname{atan}{\left(x^{8} \right)}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\frac{\tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)} \operatorname{atan}{\left(x^{8} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{\tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)}}}{\frac{d}{d x} \operatorname{atan}{\left(x^{8} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(x^{16} + 1\right) \left(\frac{14 x \tan^{8}{\left(12 x \right)}}{\sqrt{1 - 49 x^{4}} \sin^{2}{\left(8 x \right)}} + \frac{\left(96 \tan^{2}{\left(12 x \right)} + 96\right) \tan^{7}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)}} - \frac{16 \cos{\left(8 x \right)} \tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{3}{\left(8 x \right)}}\right)}{8 x^{7}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{14 x \tan^{8}{\left(12 x \right)}}{\sqrt{1 - 49 x^{4}} \sin^{2}{\left(8 x \right)}} + \frac{96 \tan^{9}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)}} + \frac{96 \tan^{7}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)}} - \frac{16 \cos{\left(8 x \right)} \tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{3}{\left(8 x \right)}}}{8 x^{7}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{14 x \tan^{8}{\left(12 x \right)}}{\sqrt{1 - 49 x^{4}} \sin^{2}{\left(8 x \right)}} + \frac{96 \tan^{9}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)}} + \frac{96 \tan^{7}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)}} - \frac{16 \cos{\left(8 x \right)} \tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{3}{\left(8 x \right)}}}{8 x^{7}}\right)$$
=
$$47029248$$
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]
/ 8 / 2\\
|tan (12*x)*asin\7*x /|
lim |---------------------|
x->0+| / 8\ 2 |
\ atan\x /*sin (8*x) /
$$\lim_{x \to 0^+}\left(\frac{\tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)} \operatorname{atan}{\left(x^{8} \right)}}\right)$$
47029248
$$47029248$$
= 47029248
/ 8 / 2\\
|tan (12*x)*asin\7*x /|
lim |---------------------|
x->0-| / 8\ 2 |
\ atan\x /*sin (8*x) /
$$\lim_{x \to 0^-}\left(\frac{\tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)} \operatorname{atan}{\left(x^{8} \right)}}\right)$$
47029248
$$47029248$$
= 47029248
= 47029248
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)} \operatorname{atan}{\left(x^{8} \right)}}\right) = 47029248$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)} \operatorname{atan}{\left(x^{8} \right)}}\right) = 47029248$$
$$\lim_{x \to \infty}\left(\frac{\tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)} \operatorname{atan}{\left(x^{8} \right)}}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)} \operatorname{atan}{\left(x^{8} \right)}}\right) = \frac{4 \tan^{8}{\left(12 \right)} \operatorname{asin}{\left(7 \right)}}{\pi \sin^{2}{\left(8 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)} \operatorname{atan}{\left(x^{8} \right)}}\right) = \frac{4 \tan^{8}{\left(12 \right)} \operatorname{asin}{\left(7 \right)}}{\pi \sin^{2}{\left(8 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)} \operatorname{atan}{\left(x^{8} \right)}}\right)$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)} \operatorname{atan}{\left(x^{8} \right)}}\right) = 47029248$$
$$\lim_{x \to \infty}\left(\frac{\tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)} \operatorname{atan}{\left(x^{8} \right)}}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)} \operatorname{atan}{\left(x^{8} \right)}}\right) = \frac{4 \tan^{8}{\left(12 \right)} \operatorname{asin}{\left(7 \right)}}{\pi \sin^{2}{\left(8 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)} \operatorname{atan}{\left(x^{8} \right)}}\right) = \frac{4 \tan^{8}{\left(12 \right)} \operatorname{asin}{\left(7 \right)}}{\pi \sin^{2}{\left(8 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\tan^{8}{\left(12 x \right)} \operatorname{asin}{\left(7 x^{2} \right)}}{\sin^{2}{\left(8 x \right)} \operatorname{atan}{\left(x^{8} \right)}}\right)$$
Más detalles con x→-oo