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Otras calculadoras:
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- Límite de la función:
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Límite de x/(-2+x)
-
Límite de x^(-2)
-
Límite de (-4+x^2)/(6+x^2-5*x)
-
Límite de (2-sqrt(-3+x))/(-49+x^2)
-
Expresiones idénticas
- (pi+ dos *x)*tan(x)
- ( número pi más 2 multiplicar por x) multiplicar por tangente de (x)
- ( número pi más dos multiplicar por x) multiplicar por tangente de (x)
- (pi+2x)tan(x)
- pi+2xtanx
-
Expresiones semejantes
- x^(-pi+2*x)*tan(x)
- (pi-2*x)*tan(x)
-
Expresiones con funciones
- Número Pi pi
- pi/(x*cot(pi*x/2))
- pi*n*x*sin(-3/2+x/2)/(6*cot(a))
- pi/atan(x)
- Piecewise((6-3*x,x<2),(6-2*x^2,True))
- pi^2*x^2*(-2+x)^2/2
- Tangente tan
- tan(x)/tan(3*x)
- tan(3*x)/(2*sin(x))
- tan(5*x)/sin(2*x)
- tan(4*x)/sin(x)
- tan(k*x)/x
Límite de la función (pi+2*x)*tan(x)
Solución
Ha introducido
[src]
lim ((pi + 2*x)*tan(x))
-pi
x->----+
2
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^+}\left(\left(2 x + \pi\right) \tan{\left(x \right)}\right)$$
Limit((pi + 2*x)*tan(x), x, (-pi)/2)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^+}\left(2 x + \pi\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^+} \frac{1}{\tan{\left(x \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^+}\left(\left(2 x + \pi\right) \tan{\left(x \right)}\right)$$
=
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^+}\left(\frac{\frac{d}{d x} \left(2 x + \pi\right)}{\frac{d}{d x} \frac{1}{\tan{\left(x \right)}}}\right)$$
=
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^+}\left(\frac{2 \tan^{2}{\left(x \right)}}{- \tan^{2}{\left(x \right)} - 1}\right)$$
=
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{1}{- \tan^{2}{\left(x \right)} - 1}}{\frac{d}{d x} \frac{1}{2 \tan^{2}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^+}\left(\frac{1}{\left(- \frac{2 \tan^{2}{\left(x \right)}}{4 \tan^{6}{\left(x \right)} + 4 \tan^{4}{\left(x \right)}} - \frac{2}{4 \tan^{6}{\left(x \right)} + 4 \tan^{4}{\left(x \right)}}\right) \left(\tan^{4}{\left(x \right)} + 2 \tan^{2}{\left(x \right)} + 1\right)}\right)$$
=
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{1}{\tan^{4}{\left(x \right)} + 2 \tan^{2}{\left(x \right)} + 1}}{\frac{d}{d x} \left(- \frac{2 \tan^{2}{\left(x \right)}}{4 \tan^{6}{\left(x \right)} + 4 \tan^{4}{\left(x \right)}} - \frac{2}{4 \tan^{6}{\left(x \right)} + 4 \tan^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^+}\left(\frac{- 2 \left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} - \left(4 \tan^{2}{\left(x \right)} + 4\right) \tan^{3}{\left(x \right)}}{\left(- \frac{2 \left(- 4 \left(4 \tan^{2}{\left(x \right)} + 4\right) \tan^{3}{\left(x \right)} - 4 \left(6 \tan^{2}{\left(x \right)} + 6\right) \tan^{5}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\left(4 \tan^{6}{\left(x \right)} + 4 \tan^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(- 4 \left(4 \tan^{2}{\left(x \right)} + 4\right) \tan^{3}{\left(x \right)} - 4 \left(6 \tan^{2}{\left(x \right)} + 6\right) \tan^{5}{\left(x \right)}\right)}{\left(4 \tan^{6}{\left(x \right)} + 4 \tan^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}}{4 \tan^{6}{\left(x \right)} + 4 \tan^{4}{\left(x \right)}}\right) \left(\tan^{4}{\left(x \right)} + 2 \tan^{2}{\left(x \right)} + 1\right)^{2}}\right)$$
=
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^+}\left(\frac{- 2 \left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} - \left(4 \tan^{2}{\left(x \right)} + 4\right) \tan^{3}{\left(x \right)}}{\left(- \frac{2 \left(- 4 \left(4 \tan^{2}{\left(x \right)} + 4\right) \tan^{3}{\left(x \right)} - 4 \left(6 \tan^{2}{\left(x \right)} + 6\right) \tan^{5}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\left(4 \tan^{6}{\left(x \right)} + 4 \tan^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(- 4 \left(4 \tan^{2}{\left(x \right)} + 4\right) \tan^{3}{\left(x \right)} - 4 \left(6 \tan^{2}{\left(x \right)} + 6\right) \tan^{5}{\left(x \right)}\right)}{\left(4 \tan^{6}{\left(x \right)} + 4 \tan^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}}{4 \tan^{6}{\left(x \right)} + 4 \tan^{4}{\left(x \right)}}\right) \left(\tan^{4}{\left(x \right)} + 2 \tan^{2}{\left(x \right)} + 1\right)^{2}}\right)$$
=
$$-2$$
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
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^+}\left(2 x + \pi\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^+} \frac{1}{\tan{\left(x \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^+}\left(\left(2 x + \pi\right) \tan{\left(x \right)}\right)$$
=
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^+}\left(\frac{\frac{d}{d x} \left(2 x + \pi\right)}{\frac{d}{d x} \frac{1}{\tan{\left(x \right)}}}\right)$$
=
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^+}\left(\frac{2 \tan^{2}{\left(x \right)}}{- \tan^{2}{\left(x \right)} - 1}\right)$$
=
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{1}{- \tan^{2}{\left(x \right)} - 1}}{\frac{d}{d x} \frac{1}{2 \tan^{2}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^+}\left(\frac{1}{\left(- \frac{2 \tan^{2}{\left(x \right)}}{4 \tan^{6}{\left(x \right)} + 4 \tan^{4}{\left(x \right)}} - \frac{2}{4 \tan^{6}{\left(x \right)} + 4 \tan^{4}{\left(x \right)}}\right) \left(\tan^{4}{\left(x \right)} + 2 \tan^{2}{\left(x \right)} + 1\right)}\right)$$
=
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{1}{\tan^{4}{\left(x \right)} + 2 \tan^{2}{\left(x \right)} + 1}}{\frac{d}{d x} \left(- \frac{2 \tan^{2}{\left(x \right)}}{4 \tan^{6}{\left(x \right)} + 4 \tan^{4}{\left(x \right)}} - \frac{2}{4 \tan^{6}{\left(x \right)} + 4 \tan^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^+}\left(\frac{- 2 \left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} - \left(4 \tan^{2}{\left(x \right)} + 4\right) \tan^{3}{\left(x \right)}}{\left(- \frac{2 \left(- 4 \left(4 \tan^{2}{\left(x \right)} + 4\right) \tan^{3}{\left(x \right)} - 4 \left(6 \tan^{2}{\left(x \right)} + 6\right) \tan^{5}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\left(4 \tan^{6}{\left(x \right)} + 4 \tan^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(- 4 \left(4 \tan^{2}{\left(x \right)} + 4\right) \tan^{3}{\left(x \right)} - 4 \left(6 \tan^{2}{\left(x \right)} + 6\right) \tan^{5}{\left(x \right)}\right)}{\left(4 \tan^{6}{\left(x \right)} + 4 \tan^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}}{4 \tan^{6}{\left(x \right)} + 4 \tan^{4}{\left(x \right)}}\right) \left(\tan^{4}{\left(x \right)} + 2 \tan^{2}{\left(x \right)} + 1\right)^{2}}\right)$$
=
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^+}\left(\frac{- 2 \left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} - \left(4 \tan^{2}{\left(x \right)} + 4\right) \tan^{3}{\left(x \right)}}{\left(- \frac{2 \left(- 4 \left(4 \tan^{2}{\left(x \right)} + 4\right) \tan^{3}{\left(x \right)} - 4 \left(6 \tan^{2}{\left(x \right)} + 6\right) \tan^{5}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\left(4 \tan^{6}{\left(x \right)} + 4 \tan^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(- 4 \left(4 \tan^{2}{\left(x \right)} + 4\right) \tan^{3}{\left(x \right)} - 4 \left(6 \tan^{2}{\left(x \right)} + 6\right) \tan^{5}{\left(x \right)}\right)}{\left(4 \tan^{6}{\left(x \right)} + 4 \tan^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}}{4 \tan^{6}{\left(x \right)} + 4 \tan^{4}{\left(x \right)}}\right) \left(\tan^{4}{\left(x \right)} + 2 \tan^{2}{\left(x \right)} + 1\right)^{2}}\right)$$
=
$$-2$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)
Gráfica
A la izquierda y a la derecha
[src]
lim ((pi + 2*x)*tan(x))
-pi
x->----+
2
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^+}\left(\left(2 x + \pi\right) \tan{\left(x \right)}\right)$$
-2
$$-2$$
= -2
lim ((pi + 2*x)*tan(x))
-pi
x->-----
2
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^-}\left(\left(2 x + \pi\right) \tan{\left(x \right)}\right)$$
-2
$$-2$$
= -2
= -2
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^-}\left(\left(2 x + \pi\right) \tan{\left(x \right)}\right) = -2$$
Más detalles con x→(-pi)/2 a la izquierda
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^+}\left(\left(2 x + \pi\right) \tan{\left(x \right)}\right) = -2$$
$$\lim_{x \to \infty}\left(\left(2 x + \pi\right) \tan{\left(x \right)}\right)$$
Más detalles con x→oo
$$\lim_{x \to 0^-}\left(\left(2 x + \pi\right) \tan{\left(x \right)}\right) = 0$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\left(2 x + \pi\right) \tan{\left(x \right)}\right) = 0$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\left(2 x + \pi\right) \tan{\left(x \right)}\right) = 2 \tan{\left(1 \right)} + \pi \tan{\left(1 \right)}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\left(2 x + \pi\right) \tan{\left(x \right)}\right) = 2 \tan{\left(1 \right)} + \pi \tan{\left(1 \right)}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\left(2 x + \pi\right) \tan{\left(x \right)}\right)$$
Más detalles con x→-oo
Más detalles con x→(-pi)/2 a la izquierda
$$\lim_{x \to \frac{\left(-1\right) \pi}{2}^+}\left(\left(2 x + \pi\right) \tan{\left(x \right)}\right) = -2$$
$$\lim_{x \to \infty}\left(\left(2 x + \pi\right) \tan{\left(x \right)}\right)$$
Más detalles con x→oo
$$\lim_{x \to 0^-}\left(\left(2 x + \pi\right) \tan{\left(x \right)}\right) = 0$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\left(2 x + \pi\right) \tan{\left(x \right)}\right) = 0$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\left(2 x + \pi\right) \tan{\left(x \right)}\right) = 2 \tan{\left(1 \right)} + \pi \tan{\left(1 \right)}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\left(2 x + \pi\right) \tan{\left(x \right)}\right) = 2 \tan{\left(1 \right)} + \pi \tan{\left(1 \right)}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\left(2 x + \pi\right) \tan{\left(x \right)}\right)$$
Más detalles con x→-oo