Sr Examen
Otras calculadoras:
-
¿Cómo usar?
- Límite de la función:
-
Límite de 4-3*x+2*x^2
-
Límite de ((3+x)/(-2+x))^x
-
Límite de (-8+x^3)/(-6+x+x^2)
-
Límite de (-3+sqrt(1+2*x))/(sqrt(-2+x)-sqrt(2))
-
Expresiones idénticas
- (pi/ dos -x)^ dos *tan(x)
- ( número pi dividir por 2 menos x) al cuadrado multiplicar por tangente de (x)
- ( número pi dividir por dos menos x) en el grado dos multiplicar por tangente de (x)
- (pi/2-x)2*tan(x)
- pi/2-x2*tanx
- (pi/2-x)²*tan(x)
- (pi/2-x) en el grado 2*tan(x)
- (pi/2-x)^2tan(x)
- (pi/2-x)2tan(x)
- pi/2-x2tanx
- pi/2-x^2tanx
- (pi dividir por 2-x)^2*tan(x)
-
Expresiones semejantes
- (pi/2+x)^2*tan(x)
-
Expresiones con funciones
- Número Pi pi
- pi/cos(x)-2*x*tan(x)
- Piecewise((5-x^2,x>3),(3+x,True))
- pi*(9-x^2)/sin(pi*x)
- pi-sqrt(x)-2*sqrt(x)*atan(x)
- Piecewise((4+x^2,x>=-1),(6+2*x,True))
- Tangente tan
- tan(2*x)/(2*x^2)
- tan(pi*x/2)/log(1-x)
- tan(5*x)/(3*x)
- tan(m*x)/sin(n*x)
- tan(3*x)/(7*x)
Límite de la función (pi/2-x)^2*tan(x)
Solución
Ha introducido
[src]
/ 2 \
|/pi \ |
lim ||-- - x| *tan(x)|
pi \\2 / /
x->--+
2
$$\lim_{x \to \frac{\pi}{2}^+}\left(\left(- x + \frac{\pi}{2}\right)^{2} \tan{\left(x \right)}\right)$$
Limit((pi/2 - x)^2*tan(x), x, pi/2)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to \frac{\pi}{2}^+}\left(4 x^{2} - 4 \pi x + \pi^{2}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{4}{\tan{\left(x \right)}}\right) = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \frac{\pi}{2}^+}\left(\left(- x + \frac{\pi}{2}\right)^{2} \tan{\left(x \right)}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\left(\pi - 2 x\right)^{2} \tan{\left(x \right)}}{4}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \left(4 x^{2} - 4 \pi x + \pi^{2}\right)}{\frac{d}{d x} \frac{4}{\tan{\left(x \right)}}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{2 x \tan^{2}{\left(x \right)} - \pi \tan^{2}{\left(x \right)}}{- \tan^{2}{\left(x \right)} - 1}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{1}{- \tan^{2}{\left(x \right)} - 1}}{\frac{d}{d x} \frac{4}{8 x \tan^{2}{\left(x \right)} - 4 \pi \tan^{2}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{1}{\left(\frac{4 \tan^{4}{\left(x \right)}}{128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}} + \frac{8 \tan^{2}{\left(x \right)}}{128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}} + \frac{4}{128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}}\right) \left(- 16 x \tan^{3}{\left(x \right)} - 16 x \tan{\left(x \right)} + 8 \pi \tan^{3}{\left(x \right)} - 8 \tan^{2}{\left(x \right)} + 8 \pi \tan{\left(x \right)}\right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{1}{- 16 x \tan^{3}{\left(x \right)} - 16 x \tan{\left(x \right)} + 8 \pi \tan^{3}{\left(x \right)} - 8 \tan^{2}{\left(x \right)} + 8 \pi \tan{\left(x \right)}}}{\frac{d}{d x} \left(\frac{4 \tan^{4}{\left(x \right)}}{128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}} + \frac{8 \tan^{2}{\left(x \right)}}{128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}} + \frac{4}{128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{16 x \left(\tan^{2}{\left(x \right)} + 1\right) + 16 x \left(3 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)} - 8 \pi \left(\tan^{2}{\left(x \right)} + 1\right) + 8 \left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} - 8 \pi \left(3 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)} + 16 \tan^{3}{\left(x \right)} + 16 \tan{\left(x \right)}}{\left(- 16 x \tan^{3}{\left(x \right)} - 16 x \tan{\left(x \right)} + 8 \pi \tan^{3}{\left(x \right)} - 8 \tan^{2}{\left(x \right)} + 8 \pi \tan{\left(x \right)}\right)^{2} \left(\frac{8 \left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}}{128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}} + \frac{4 \left(4 \tan^{2}{\left(x \right)} + 4\right) \tan^{3}{\left(x \right)}}{128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}} + \frac{4 \left(- 128 x^{2} \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} - 128 x^{2} \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} + 128 \pi x \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} + 128 \pi x \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} - 256 x \tan^{7}{\left(x \right)} - 256 x \tan^{5}{\left(x \right)} - 32 \pi^{2} \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} - 32 \pi^{2} \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} + 128 \pi \tan^{7}{\left(x \right)} + 128 \pi \tan^{5}{\left(x \right)}\right) \tan^{4}{\left(x \right)}}{\left(128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}\right)^{2}} + \frac{8 \left(- 128 x^{2} \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} - 128 x^{2} \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} + 128 \pi x \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} + 128 \pi x \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} - 256 x \tan^{7}{\left(x \right)} - 256 x \tan^{5}{\left(x \right)} - 32 \pi^{2} \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} - 32 \pi^{2} \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} + 128 \pi \tan^{7}{\left(x \right)} + 128 \pi \tan^{5}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\left(128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}\right)^{2}} + \frac{4 \left(- 128 x^{2} \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} - 128 x^{2} \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} + 128 \pi x \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} + 128 \pi x \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} - 256 x \tan^{7}{\left(x \right)} - 256 x \tan^{5}{\left(x \right)} - 32 \pi^{2} \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} - 32 \pi^{2} \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} + 128 \pi \tan^{7}{\left(x \right)} + 128 \pi \tan^{5}{\left(x \right)}\right)}{\left(128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}\right)^{2}}\right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{16 x \left(\tan^{2}{\left(x \right)} + 1\right) + 16 x \left(3 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)} - 8 \pi \left(\tan^{2}{\left(x \right)} + 1\right) + 8 \left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} - 8 \pi \left(3 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)} + 16 \tan^{3}{\left(x \right)} + 16 \tan{\left(x \right)}}{\left(- 16 x \tan^{3}{\left(x \right)} - 16 x \tan{\left(x \right)} + 8 \pi \tan^{3}{\left(x \right)} - 8 \tan^{2}{\left(x \right)} + 8 \pi \tan{\left(x \right)}\right)^{2} \left(\frac{8 \left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}}{128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}} + \frac{4 \left(4 \tan^{2}{\left(x \right)} + 4\right) \tan^{3}{\left(x \right)}}{128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}} + \frac{4 \left(- 128 x^{2} \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} - 128 x^{2} \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} + 128 \pi x \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} + 128 \pi x \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} - 256 x \tan^{7}{\left(x \right)} - 256 x \tan^{5}{\left(x \right)} - 32 \pi^{2} \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} - 32 \pi^{2} \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} + 128 \pi \tan^{7}{\left(x \right)} + 128 \pi \tan^{5}{\left(x \right)}\right) \tan^{4}{\left(x \right)}}{\left(128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}\right)^{2}} + \frac{8 \left(- 128 x^{2} \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} - 128 x^{2} \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} + 128 \pi x \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} + 128 \pi x \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} - 256 x \tan^{7}{\left(x \right)} - 256 x \tan^{5}{\left(x \right)} - 32 \pi^{2} \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} - 32 \pi^{2} \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} + 128 \pi \tan^{7}{\left(x \right)} + 128 \pi \tan^{5}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\left(128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}\right)^{2}} + \frac{4 \left(- 128 x^{2} \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} - 128 x^{2} \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} + 128 \pi x \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} + 128 \pi x \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} - 256 x \tan^{7}{\left(x \right)} - 256 x \tan^{5}{\left(x \right)} - 32 \pi^{2} \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} - 32 \pi^{2} \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} + 128 \pi \tan^{7}{\left(x \right)} + 128 \pi \tan^{5}{\left(x \right)}\right)}{\left(128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}\right)^{2}}\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
$$\lim_{x \to \frac{\pi}{2}^+}\left(4 x^{2} - 4 \pi x + \pi^{2}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{4}{\tan{\left(x \right)}}\right) = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \frac{\pi}{2}^+}\left(\left(- x + \frac{\pi}{2}\right)^{2} \tan{\left(x \right)}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\left(\pi - 2 x\right)^{2} \tan{\left(x \right)}}{4}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \left(4 x^{2} - 4 \pi x + \pi^{2}\right)}{\frac{d}{d x} \frac{4}{\tan{\left(x \right)}}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{2 x \tan^{2}{\left(x \right)} - \pi \tan^{2}{\left(x \right)}}{- \tan^{2}{\left(x \right)} - 1}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{1}{- \tan^{2}{\left(x \right)} - 1}}{\frac{d}{d x} \frac{4}{8 x \tan^{2}{\left(x \right)} - 4 \pi \tan^{2}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{1}{\left(\frac{4 \tan^{4}{\left(x \right)}}{128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}} + \frac{8 \tan^{2}{\left(x \right)}}{128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}} + \frac{4}{128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}}\right) \left(- 16 x \tan^{3}{\left(x \right)} - 16 x \tan{\left(x \right)} + 8 \pi \tan^{3}{\left(x \right)} - 8 \tan^{2}{\left(x \right)} + 8 \pi \tan{\left(x \right)}\right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{1}{- 16 x \tan^{3}{\left(x \right)} - 16 x \tan{\left(x \right)} + 8 \pi \tan^{3}{\left(x \right)} - 8 \tan^{2}{\left(x \right)} + 8 \pi \tan{\left(x \right)}}}{\frac{d}{d x} \left(\frac{4 \tan^{4}{\left(x \right)}}{128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}} + \frac{8 \tan^{2}{\left(x \right)}}{128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}} + \frac{4}{128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{16 x \left(\tan^{2}{\left(x \right)} + 1\right) + 16 x \left(3 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)} - 8 \pi \left(\tan^{2}{\left(x \right)} + 1\right) + 8 \left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} - 8 \pi \left(3 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)} + 16 \tan^{3}{\left(x \right)} + 16 \tan{\left(x \right)}}{\left(- 16 x \tan^{3}{\left(x \right)} - 16 x \tan{\left(x \right)} + 8 \pi \tan^{3}{\left(x \right)} - 8 \tan^{2}{\left(x \right)} + 8 \pi \tan{\left(x \right)}\right)^{2} \left(\frac{8 \left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}}{128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}} + \frac{4 \left(4 \tan^{2}{\left(x \right)} + 4\right) \tan^{3}{\left(x \right)}}{128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}} + \frac{4 \left(- 128 x^{2} \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} - 128 x^{2} \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} + 128 \pi x \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} + 128 \pi x \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} - 256 x \tan^{7}{\left(x \right)} - 256 x \tan^{5}{\left(x \right)} - 32 \pi^{2} \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} - 32 \pi^{2} \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} + 128 \pi \tan^{7}{\left(x \right)} + 128 \pi \tan^{5}{\left(x \right)}\right) \tan^{4}{\left(x \right)}}{\left(128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}\right)^{2}} + \frac{8 \left(- 128 x^{2} \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} - 128 x^{2} \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} + 128 \pi x \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} + 128 \pi x \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} - 256 x \tan^{7}{\left(x \right)} - 256 x \tan^{5}{\left(x \right)} - 32 \pi^{2} \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} - 32 \pi^{2} \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} + 128 \pi \tan^{7}{\left(x \right)} + 128 \pi \tan^{5}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\left(128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}\right)^{2}} + \frac{4 \left(- 128 x^{2} \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} - 128 x^{2} \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} + 128 \pi x \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} + 128 \pi x \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} - 256 x \tan^{7}{\left(x \right)} - 256 x \tan^{5}{\left(x \right)} - 32 \pi^{2} \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} - 32 \pi^{2} \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} + 128 \pi \tan^{7}{\left(x \right)} + 128 \pi \tan^{5}{\left(x \right)}\right)}{\left(128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}\right)^{2}}\right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{16 x \left(\tan^{2}{\left(x \right)} + 1\right) + 16 x \left(3 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)} - 8 \pi \left(\tan^{2}{\left(x \right)} + 1\right) + 8 \left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} - 8 \pi \left(3 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)} + 16 \tan^{3}{\left(x \right)} + 16 \tan{\left(x \right)}}{\left(- 16 x \tan^{3}{\left(x \right)} - 16 x \tan{\left(x \right)} + 8 \pi \tan^{3}{\left(x \right)} - 8 \tan^{2}{\left(x \right)} + 8 \pi \tan{\left(x \right)}\right)^{2} \left(\frac{8 \left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}}{128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}} + \frac{4 \left(4 \tan^{2}{\left(x \right)} + 4\right) \tan^{3}{\left(x \right)}}{128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}} + \frac{4 \left(- 128 x^{2} \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} - 128 x^{2} \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} + 128 \pi x \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} + 128 \pi x \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} - 256 x \tan^{7}{\left(x \right)} - 256 x \tan^{5}{\left(x \right)} - 32 \pi^{2} \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} - 32 \pi^{2} \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} + 128 \pi \tan^{7}{\left(x \right)} + 128 \pi \tan^{5}{\left(x \right)}\right) \tan^{4}{\left(x \right)}}{\left(128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}\right)^{2}} + \frac{8 \left(- 128 x^{2} \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} - 128 x^{2} \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} + 128 \pi x \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} + 128 \pi x \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} - 256 x \tan^{7}{\left(x \right)} - 256 x \tan^{5}{\left(x \right)} - 32 \pi^{2} \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} - 32 \pi^{2} \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} + 128 \pi \tan^{7}{\left(x \right)} + 128 \pi \tan^{5}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\left(128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}\right)^{2}} + \frac{4 \left(- 128 x^{2} \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} - 128 x^{2} \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} + 128 \pi x \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} + 128 \pi x \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} - 256 x \tan^{7}{\left(x \right)} - 256 x \tan^{5}{\left(x \right)} - 32 \pi^{2} \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} - 32 \pi^{2} \left(7 \tan^{2}{\left(x \right)} + 7\right) \tan^{6}{\left(x \right)} + 128 \pi \tan^{7}{\left(x \right)} + 128 \pi \tan^{5}{\left(x \right)}\right)}{\left(128 x^{2} \tan^{7}{\left(x \right)} + 128 x^{2} \tan^{5}{\left(x \right)} - 128 \pi x \tan^{7}{\left(x \right)} - 128 \pi x \tan^{5}{\left(x \right)} + 32 \pi^{2} \tan^{7}{\left(x \right)} + 32 \pi^{2} \tan^{5}{\left(x \right)}\right)^{2}}\right)}\right)$$
=
$$0$$
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]
/ 2 \
|/pi \ |
lim ||-- - x| *tan(x)|
pi \\2 / /
x->--+
2
$$\lim_{x \to \frac{\pi}{2}^+}\left(\left(- x + \frac{\pi}{2}\right)^{2} \tan{\left(x \right)}\right)$$
0
$$0$$
= 6.12323399573679e-17
/ 2 \
|/pi \ |
lim ||-- - x| *tan(x)|
pi \\2 / /
x->---
2
$$\lim_{x \to \frac{\pi}{2}^-}\left(\left(- x + \frac{\pi}{2}\right)^{2} \tan{\left(x \right)}\right)$$
0
$$0$$
= 6.12323399573674e-17
= 6.12323399573674e-17
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to \frac{\pi}{2}^-}\left(\left(- x + \frac{\pi}{2}\right)^{2} \tan{\left(x \right)}\right) = 0$$
Más detalles con x→pi/2 a la izquierda
$$\lim_{x \to \frac{\pi}{2}^+}\left(\left(- x + \frac{\pi}{2}\right)^{2} \tan{\left(x \right)}\right) = 0$$
$$\lim_{x \to \infty}\left(\left(- x + \frac{\pi}{2}\right)^{2} \tan{\left(x \right)}\right)$$
Más detalles con x→oo
$$\lim_{x \to 0^-}\left(\left(- x + \frac{\pi}{2}\right)^{2} \tan{\left(x \right)}\right) = 0$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\left(- x + \frac{\pi}{2}\right)^{2} \tan{\left(x \right)}\right) = 0$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\left(- x + \frac{\pi}{2}\right)^{2} \tan{\left(x \right)}\right) = - \pi \tan{\left(1 \right)} + \tan{\left(1 \right)} + \frac{\pi^{2} \tan{\left(1 \right)}}{4}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\left(- x + \frac{\pi}{2}\right)^{2} \tan{\left(x \right)}\right) = - \pi \tan{\left(1 \right)} + \tan{\left(1 \right)} + \frac{\pi^{2} \tan{\left(1 \right)}}{4}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\left(- x + \frac{\pi}{2}\right)^{2} \tan{\left(x \right)}\right)$$
Más detalles con x→-oo
Más detalles con x→pi/2 a la izquierda
$$\lim_{x \to \frac{\pi}{2}^+}\left(\left(- x + \frac{\pi}{2}\right)^{2} \tan{\left(x \right)}\right) = 0$$
$$\lim_{x \to \infty}\left(\left(- x + \frac{\pi}{2}\right)^{2} \tan{\left(x \right)}\right)$$
Más detalles con x→oo
$$\lim_{x \to 0^-}\left(\left(- x + \frac{\pi}{2}\right)^{2} \tan{\left(x \right)}\right) = 0$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\left(- x + \frac{\pi}{2}\right)^{2} \tan{\left(x \right)}\right) = 0$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\left(- x + \frac{\pi}{2}\right)^{2} \tan{\left(x \right)}\right) = - \pi \tan{\left(1 \right)} + \tan{\left(1 \right)} + \frac{\pi^{2} \tan{\left(1 \right)}}{4}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\left(- x + \frac{\pi}{2}\right)^{2} \tan{\left(x \right)}\right) = - \pi \tan{\left(1 \right)} + \tan{\left(1 \right)} + \frac{\pi^{2} \tan{\left(1 \right)}}{4}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\left(- x + \frac{\pi}{2}\right)^{2} \tan{\left(x \right)}\right)$$
Más detalles con x→-oo