Sr Examen
Otras calculadoras:
-
¿Cómo usar?
- Límite de la función:
-
Límite de (6+x^2-5*x)/(20+x^2-12*x)
-
Límite de tan(5*x)/x
-
Límite de (3-sqrt(5+x))/(1-sqrt(5-x))
-
Límite de x^2
-
Expresiones idénticas
- (x-pi)*cot(x)
- (x menos número pi ) multiplicar por cotangente de (x)
- (x-pi)cot(x)
- x-picotx
-
Expresiones semejantes
- (x+pi)*cot(x)
-
Expresiones con funciones
- Número Pi pi
- pi/(2*x)
- Piecewise((1+5*x,x>=1),(-2+x^2,True))
- Piecewise((4+x,x<-1),(2+x^2,x<=1),(2*x,x>=1))
- pi/(x*cot(pi*x/2))
- pi/4
- Cotangente cot
- cot(5*x)*sin(7*x)
- coth(x)
- cot(8*x)
- cot(5*x)*sin(3*x)
- cot(2*x)*sin(6*x)
Límite de la función (x-pi)*cot(x)
Solución
Ha introducido
[src]
lim ((x - pi)*cot(x)) x->pi+
$$\lim_{x \to \pi^+}\left(\left(x - \pi\right) \cot{\left(x \right)}\right)$$
Limit((x - pi)*cot(x), x, pi)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to \pi^+}\left(x - \pi\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to \pi^+} \frac{1}{\cot{\left(x \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \pi^+}\left(\left(x - \pi\right) \cot{\left(x \right)}\right)$$
=
$$\lim_{x \to \pi^+}\left(\frac{\frac{d}{d x} \left(x - \pi\right)}{\frac{d}{d x} \frac{1}{\cot{\left(x \right)}}}\right)$$
=
$$\lim_{x \to \pi^+}\left(\frac{\cot^{2}{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right)$$
=
$$\lim_{x \to \pi^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot^{2}{\left(x \right)} + 1}}{\frac{d}{d x} \frac{1}{\cot^{2}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to \pi^+}\left(\frac{1}{\left(2 \cot^{2}{\left(x \right)} + 2\right) \left(- \frac{\cot^{4}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{2 \cot^{2}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{1}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to \pi^+}\left(\frac{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(x \right)} + 2}}{\frac{d}{d x} \left(- \frac{\cot^{4}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{2 \cot^{2}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{1}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to \pi^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{\left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{\left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to \pi^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{\left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{\left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$1$$
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
$$\lim_{x \to \pi^+}\left(x - \pi\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to \pi^+} \frac{1}{\cot{\left(x \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \pi^+}\left(\left(x - \pi\right) \cot{\left(x \right)}\right)$$
=
$$\lim_{x \to \pi^+}\left(\frac{\frac{d}{d x} \left(x - \pi\right)}{\frac{d}{d x} \frac{1}{\cot{\left(x \right)}}}\right)$$
=
$$\lim_{x \to \pi^+}\left(\frac{\cot^{2}{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right)$$
=
$$\lim_{x \to \pi^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot^{2}{\left(x \right)} + 1}}{\frac{d}{d x} \frac{1}{\cot^{2}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to \pi^+}\left(\frac{1}{\left(2 \cot^{2}{\left(x \right)} + 2\right) \left(- \frac{\cot^{4}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{2 \cot^{2}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{1}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to \pi^+}\left(\frac{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(x \right)} + 2}}{\frac{d}{d x} \left(- \frac{\cot^{4}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{2 \cot^{2}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{1}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to \pi^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{\left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{\left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to \pi^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{\left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{\left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$1$$
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 ((x - pi)*cot(x)) x->pi+
$$\lim_{x \to \pi^+}\left(\left(x - \pi\right) \cot{\left(x \right)}\right)$$
1
$$1$$
= 1.0
lim ((x - pi)*cot(x)) x->pi-
$$\lim_{x \to \pi^-}\left(\left(x - \pi\right) \cot{\left(x \right)}\right)$$
1
$$1$$
= 1.0
= 1.0
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to \pi^-}\left(\left(x - \pi\right) \cot{\left(x \right)}\right) = 1$$
Más detalles con x→pi a la izquierda
$$\lim_{x \to \pi^+}\left(\left(x - \pi\right) \cot{\left(x \right)}\right) = 1$$
$$\lim_{x \to \infty}\left(\left(x - \pi\right) \cot{\left(x \right)}\right)$$
Más detalles con x→oo
$$\lim_{x \to 0^-}\left(\left(x - \pi\right) \cot{\left(x \right)}\right) = \infty$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\left(x - \pi\right) \cot{\left(x \right)}\right) = -\infty$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\left(x - \pi\right) \cot{\left(x \right)}\right) = - \frac{-1 + \pi}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\left(x - \pi\right) \cot{\left(x \right)}\right) = - \frac{-1 + \pi}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\left(x - \pi\right) \cot{\left(x \right)}\right)$$
Más detalles con x→-oo
Más detalles con x→pi a la izquierda
$$\lim_{x \to \pi^+}\left(\left(x - \pi\right) \cot{\left(x \right)}\right) = 1$$
$$\lim_{x \to \infty}\left(\left(x - \pi\right) \cot{\left(x \right)}\right)$$
Más detalles con x→oo
$$\lim_{x \to 0^-}\left(\left(x - \pi\right) \cot{\left(x \right)}\right) = \infty$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\left(x - \pi\right) \cot{\left(x \right)}\right) = -\infty$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\left(x - \pi\right) \cot{\left(x \right)}\right) = - \frac{-1 + \pi}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\left(x - \pi\right) \cot{\left(x \right)}\right) = - \frac{-1 + \pi}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\left(x - \pi\right) \cot{\left(x \right)}\right)$$
Más detalles con x→-oo
Gráfico