Sr Examen
Otras calculadoras:
-
¿Cómo usar?
- Límite de la función:
-
Límite de (1-cos(5*x))/x^2
-
Límite de x/(-1+sqrt(1+3*x))
-
Límite de (-27+x^3)/(-9+x^2)
-
Límite de (-1-4*x+5*x^2)/(-1+x)
-
Expresiones idénticas
- (x+x^ tres -tan(diez *x)+tan(sin(nueve *x)))/x^ tres
- (x más x al cubo menos tangente de (10 multiplicar por x) más tangente de ( seno de (9 multiplicar por x))) dividir por x al cubo
- (x más x en el grado tres menos tangente de (diez multiplicar por x) más tangente de ( seno de (nueve multiplicar por x))) dividir por x en el grado tres
- (x+x3-tan(10*x)+tan(sin(9*x)))/x3
- x+x3-tan10*x+tansin9*x/x3
- (x+x³-tan(10*x)+tan(sin(9*x)))/x³
- (x+x en el grado 3-tan(10*x)+tan(sin(9*x)))/x en el grado 3
- (x+x^3-tan(10x)+tan(sin(9x)))/x^3
- (x+x3-tan(10x)+tan(sin(9x)))/x3
- x+x3-tan10x+tansin9x/x3
- x+x^3-tan10x+tansin9x/x^3
- (x+x^3-tan(10*x)+tan(sin(9*x))) dividir por x^3
-
Expresiones semejantes
- (x+x^3+tan(10*x)+tan(sin(9*x)))/x^3
- (x+x^3-tan(10*x)-tan(sin(9*x)))/x^3
- (x-x^3-tan(10*x)+tan(sin(9*x)))/x^3
-
Expresiones con funciones
- Tangente tan
- tan(8*x)/(5*x)
- tan(6*x)/(3*x)
- tan(7*x)/x
- tan(9*x)/(8*x)
- tan(x)^2/(1-cos(x))
- Tangente tan
- tan(8*x)/(5*x)
- tan(6*x)/(3*x)
- tan(7*x)/x
- tan(9*x)/(8*x)
- tan(x)^2/(1-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)
Límite de la función (x+x^3-tan(10*x)+tan(sin(9*x)))/x^3
Solución
Ha introducido
[src]
/ 3 \
|x + x - tan(10*x) + tan(sin(9*x))|
lim |----------------------------------|
x->0+| 3 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{\left(\left(x^{3} + x\right) - \tan{\left(10 x \right)}\right) + \tan{\left(\sin{\left(9 x \right)} \right)}}{x^{3}}\right)$$
Limit((x + x^3 - tan(10*x) + tan(sin(9*x)))/x^3, x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(x^{3} + x - \tan{\left(10 x \right)} + \tan{\left(\sin{\left(9 x \right)} \right)}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} x^{3} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{\left(\left(x^{3} + x\right) - \tan{\left(10 x \right)}\right) + \tan{\left(\sin{\left(9 x \right)} \right)}}{x^{3}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\frac{x \left(x^{2} + 1\right) - \tan{\left(10 x \right)} + \tan{\left(\sin{\left(9 x \right)} \right)}}{x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(x^{3} + x - \tan{\left(10 x \right)} + \tan{\left(\sin{\left(9 x \right)} \right)}\right)}{\frac{d}{d x} x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{3 x^{2} + 9 \cos{\left(9 x \right)} \tan^{2}{\left(\sin{\left(9 x \right)} \right)} + 9 \cos{\left(9 x \right)} - 10 \tan^{2}{\left(10 x \right)} - 9}{3 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(3 x^{2} + 9 \cos{\left(9 x \right)} \tan^{2}{\left(\sin{\left(9 x \right)} \right)} + 9 \cos{\left(9 x \right)} - 10 \tan^{2}{\left(10 x \right)} - 9\right)}{\frac{d}{d x} 3 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{6 x - 81 \sin{\left(9 x \right)} \tan^{2}{\left(\sin{\left(9 x \right)} \right)} - 81 \sin{\left(9 x \right)} + 162 \cos^{2}{\left(9 x \right)} \tan^{3}{\left(\sin{\left(9 x \right)} \right)} + 162 \cos^{2}{\left(9 x \right)} \tan{\left(\sin{\left(9 x \right)} \right)} - 200 \tan^{3}{\left(10 x \right)} - 200 \tan{\left(10 x \right)}}{6 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(6 x - 81 \sin{\left(9 x \right)} \tan^{2}{\left(\sin{\left(9 x \right)} \right)} - 81 \sin{\left(9 x \right)} + 162 \cos^{2}{\left(9 x \right)} \tan^{3}{\left(\sin{\left(9 x \right)} \right)} + 162 \cos^{2}{\left(9 x \right)} \tan{\left(\sin{\left(9 x \right)} \right)} - 200 \tan^{3}{\left(10 x \right)} - 200 \tan{\left(10 x \right)}\right)}{\frac{d}{d x} 6 x}\right)$$
=
$$\lim_{x \to 0^+}\left(- 729 \sin{\left(9 x \right)} \cos{\left(9 x \right)} \tan^{3}{\left(\sin{\left(9 x \right)} \right)} - 729 \sin{\left(9 x \right)} \cos{\left(9 x \right)} \tan{\left(\sin{\left(9 x \right)} \right)} + 729 \cos^{3}{\left(9 x \right)} \tan^{4}{\left(\sin{\left(9 x \right)} \right)} + 972 \cos^{3}{\left(9 x \right)} \tan^{2}{\left(\sin{\left(9 x \right)} \right)} + 243 \cos^{3}{\left(9 x \right)} - \frac{243 \cos{\left(9 x \right)} \tan^{2}{\left(\sin{\left(9 x \right)} \right)}}{2} - \frac{243 \cos{\left(9 x \right)}}{2} - 1000 \tan^{4}{\left(10 x \right)} - \frac{4000 \tan^{2}{\left(10 x \right)}}{3} - \frac{997}{3}\right)$$
=
$$\lim_{x \to 0^+}\left(- 729 \sin{\left(9 x \right)} \cos{\left(9 x \right)} \tan^{3}{\left(\sin{\left(9 x \right)} \right)} - 729 \sin{\left(9 x \right)} \cos{\left(9 x \right)} \tan{\left(\sin{\left(9 x \right)} \right)} + 729 \cos^{3}{\left(9 x \right)} \tan^{4}{\left(\sin{\left(9 x \right)} \right)} + 972 \cos^{3}{\left(9 x \right)} \tan^{2}{\left(\sin{\left(9 x \right)} \right)} + 243 \cos^{3}{\left(9 x \right)} - \frac{243 \cos{\left(9 x \right)} \tan^{2}{\left(\sin{\left(9 x \right)} \right)}}{2} - \frac{243 \cos{\left(9 x \right)}}{2} - 1000 \tan^{4}{\left(10 x \right)} - \frac{4000 \tan^{2}{\left(10 x \right)}}{3} - \frac{997}{3}\right)$$
=
$$- \frac{1265}{6}$$
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
$$\lim_{x \to 0^+}\left(x^{3} + x - \tan{\left(10 x \right)} + \tan{\left(\sin{\left(9 x \right)} \right)}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} x^{3} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{\left(\left(x^{3} + x\right) - \tan{\left(10 x \right)}\right) + \tan{\left(\sin{\left(9 x \right)} \right)}}{x^{3}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\frac{x \left(x^{2} + 1\right) - \tan{\left(10 x \right)} + \tan{\left(\sin{\left(9 x \right)} \right)}}{x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(x^{3} + x - \tan{\left(10 x \right)} + \tan{\left(\sin{\left(9 x \right)} \right)}\right)}{\frac{d}{d x} x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{3 x^{2} + 9 \cos{\left(9 x \right)} \tan^{2}{\left(\sin{\left(9 x \right)} \right)} + 9 \cos{\left(9 x \right)} - 10 \tan^{2}{\left(10 x \right)} - 9}{3 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(3 x^{2} + 9 \cos{\left(9 x \right)} \tan^{2}{\left(\sin{\left(9 x \right)} \right)} + 9 \cos{\left(9 x \right)} - 10 \tan^{2}{\left(10 x \right)} - 9\right)}{\frac{d}{d x} 3 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{6 x - 81 \sin{\left(9 x \right)} \tan^{2}{\left(\sin{\left(9 x \right)} \right)} - 81 \sin{\left(9 x \right)} + 162 \cos^{2}{\left(9 x \right)} \tan^{3}{\left(\sin{\left(9 x \right)} \right)} + 162 \cos^{2}{\left(9 x \right)} \tan{\left(\sin{\left(9 x \right)} \right)} - 200 \tan^{3}{\left(10 x \right)} - 200 \tan{\left(10 x \right)}}{6 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(6 x - 81 \sin{\left(9 x \right)} \tan^{2}{\left(\sin{\left(9 x \right)} \right)} - 81 \sin{\left(9 x \right)} + 162 \cos^{2}{\left(9 x \right)} \tan^{3}{\left(\sin{\left(9 x \right)} \right)} + 162 \cos^{2}{\left(9 x \right)} \tan{\left(\sin{\left(9 x \right)} \right)} - 200 \tan^{3}{\left(10 x \right)} - 200 \tan{\left(10 x \right)}\right)}{\frac{d}{d x} 6 x}\right)$$
=
$$\lim_{x \to 0^+}\left(- 729 \sin{\left(9 x \right)} \cos{\left(9 x \right)} \tan^{3}{\left(\sin{\left(9 x \right)} \right)} - 729 \sin{\left(9 x \right)} \cos{\left(9 x \right)} \tan{\left(\sin{\left(9 x \right)} \right)} + 729 \cos^{3}{\left(9 x \right)} \tan^{4}{\left(\sin{\left(9 x \right)} \right)} + 972 \cos^{3}{\left(9 x \right)} \tan^{2}{\left(\sin{\left(9 x \right)} \right)} + 243 \cos^{3}{\left(9 x \right)} - \frac{243 \cos{\left(9 x \right)} \tan^{2}{\left(\sin{\left(9 x \right)} \right)}}{2} - \frac{243 \cos{\left(9 x \right)}}{2} - 1000 \tan^{4}{\left(10 x \right)} - \frac{4000 \tan^{2}{\left(10 x \right)}}{3} - \frac{997}{3}\right)$$
=
$$\lim_{x \to 0^+}\left(- 729 \sin{\left(9 x \right)} \cos{\left(9 x \right)} \tan^{3}{\left(\sin{\left(9 x \right)} \right)} - 729 \sin{\left(9 x \right)} \cos{\left(9 x \right)} \tan{\left(\sin{\left(9 x \right)} \right)} + 729 \cos^{3}{\left(9 x \right)} \tan^{4}{\left(\sin{\left(9 x \right)} \right)} + 972 \cos^{3}{\left(9 x \right)} \tan^{2}{\left(\sin{\left(9 x \right)} \right)} + 243 \cos^{3}{\left(9 x \right)} - \frac{243 \cos{\left(9 x \right)} \tan^{2}{\left(\sin{\left(9 x \right)} \right)}}{2} - \frac{243 \cos{\left(9 x \right)}}{2} - 1000 \tan^{4}{\left(10 x \right)} - \frac{4000 \tan^{2}{\left(10 x \right)}}{3} - \frac{997}{3}\right)$$
=
$$- \frac{1265}{6}$$
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
$$\lim_{x \to 0^-}\left(\frac{\left(\left(x^{3} + x\right) - \tan{\left(10 x \right)}\right) + \tan{\left(\sin{\left(9 x \right)} \right)}}{x^{3}}\right) = - \frac{1265}{6}$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\left(\left(x^{3} + x\right) - \tan{\left(10 x \right)}\right) + \tan{\left(\sin{\left(9 x \right)} \right)}}{x^{3}}\right) = - \frac{1265}{6}$$
$$\lim_{x \to \infty}\left(\frac{\left(\left(x^{3} + x\right) - \tan{\left(10 x \right)}\right) + \tan{\left(\sin{\left(9 x \right)} \right)}}{x^{3}}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\left(\left(x^{3} + x\right) - \tan{\left(10 x \right)}\right) + \tan{\left(\sin{\left(9 x \right)} \right)}}{x^{3}}\right) = - \tan{\left(10 \right)} + \tan{\left(\sin{\left(9 \right)} \right)} + 2$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\left(\left(x^{3} + x\right) - \tan{\left(10 x \right)}\right) + \tan{\left(\sin{\left(9 x \right)} \right)}}{x^{3}}\right) = - \tan{\left(10 \right)} + \tan{\left(\sin{\left(9 \right)} \right)} + 2$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\left(\left(x^{3} + x\right) - \tan{\left(10 x \right)}\right) + \tan{\left(\sin{\left(9 x \right)} \right)}}{x^{3}}\right)$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\left(\left(x^{3} + x\right) - \tan{\left(10 x \right)}\right) + \tan{\left(\sin{\left(9 x \right)} \right)}}{x^{3}}\right) = - \frac{1265}{6}$$
$$\lim_{x \to \infty}\left(\frac{\left(\left(x^{3} + x\right) - \tan{\left(10 x \right)}\right) + \tan{\left(\sin{\left(9 x \right)} \right)}}{x^{3}}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\left(\left(x^{3} + x\right) - \tan{\left(10 x \right)}\right) + \tan{\left(\sin{\left(9 x \right)} \right)}}{x^{3}}\right) = - \tan{\left(10 \right)} + \tan{\left(\sin{\left(9 \right)} \right)} + 2$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\left(\left(x^{3} + x\right) - \tan{\left(10 x \right)}\right) + \tan{\left(\sin{\left(9 x \right)} \right)}}{x^{3}}\right) = - \tan{\left(10 \right)} + \tan{\left(\sin{\left(9 \right)} \right)} + 2$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\left(\left(x^{3} + x\right) - \tan{\left(10 x \right)}\right) + \tan{\left(\sin{\left(9 x \right)} \right)}}{x^{3}}\right)$$
Más detalles con x→-oo
A la izquierda y a la derecha
[src]
/ 3 \
|x + x - tan(10*x) + tan(sin(9*x))|
lim |----------------------------------|
x->0+| 3 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{\left(\left(x^{3} + x\right) - \tan{\left(10 x \right)}\right) + \tan{\left(\sin{\left(9 x \right)} \right)}}{x^{3}}\right)$$
-1265/6
$$- \frac{1265}{6}$$
= -210.833333333333
/ 3 \
|x + x - tan(10*x) + tan(sin(9*x))|
lim |----------------------------------|
x->0-| 3 |
\ x /
$$\lim_{x \to 0^-}\left(\frac{\left(\left(x^{3} + x\right) - \tan{\left(10 x \right)}\right) + \tan{\left(\sin{\left(9 x \right)} \right)}}{x^{3}}\right)$$
-1265/6
$$- \frac{1265}{6}$$
= -210.833333333333
= -210.833333333333