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Límite de (9+x^2-6*x)/(x^2-3*x)
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Límite de (-2+sqrt(-2+x))/(-6+x)
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Límite de (sqrt(12+x)-sqrt(4-x))/(-8+x^2+2*x)
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Límite de (sqrt(6+x^2-2*x)-sqrt(-6+x^2+2*x))/(3+x^2-4*x)
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Expresiones idénticas
- cot(pi*x/ seis)*log(tres -x)
- cotangente de ( número pi multiplicar por x dividir por 6) multiplicar por logaritmo de (3 menos x)
- cotangente de ( número pi multiplicar por x dividir por seis) multiplicar por logaritmo de (tres menos x)
- cot(pix/6)log(3-x)
- cotpix/6log3-x
- cot(pi*x dividir por 6)*log(3-x)
-
Expresiones semejantes
- cot(pi*x/6)*log(3+x)
-
Expresiones con funciones
- Cotangente cot
- cot(x)^(log(x)/x)
- cot(2+x)^(2+x)
- cot(z)/z^2
- cot(-6+2*x)/sin(-3+x)
- cot(x)*log(x+exp(x))
- Número Pi pi
- pi*cos(x)/3
- Piecewise((-1-x,x<-1),((1+x)^2,x<=1),(x,True))
- pi-2*asin(x/5)
- pi*x*tan(3)^2*sin(5*pi*x)*tan(2*pi*x)/cos(7*pi*x)
- pi/(2*x*(1+x)*(2+x))
- Logaritmo log
- log(cos(5*x))/log(cos(4*x))
- log(2*x)*log(-1+2*x)
- log(1+x^2-x)
- log(-3+x^2)/(2+x^2-3*x)
- log(x)/(1+2*log(x)*sin(x))
Límite de la función cot(pi*x/6)*log(3-x)
Solución
Ha introducido
[src]
/ /pi*x\ \ lim |cot|----|*log(3 - x)| x->3+\ \ 6 / /
$$\lim_{x \to 3^+}\left(\log{\left(3 - x \right)} \cot{\left(\frac{\pi x}{6} \right)}\right)$$
Limit(cot((pi*x)/6)*log(3 - x), x, 3)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 3^+} \cot{\left(\frac{\pi x}{6} \right)} = 0$$
y el límite para el denominador es
$$\lim_{x \to 3^+} \frac{1}{\log{\left(3 - x \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 3^+}\left(\log{\left(3 - x \right)} \cot{\left(\frac{\pi x}{6} \right)}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 3^+}\left(\log{\left(3 - x \right)} \cot{\left(\frac{\pi x}{6} \right)}\right)$$
=
$$\lim_{x \to 3^+}\left(\frac{\frac{d}{d x} \cot{\left(\frac{\pi x}{6} \right)}}{\frac{d}{d x} \frac{1}{\log{\left(3 - x \right)}}}\right)$$
=
$$\lim_{x \to 3^+}\left(\frac{\pi \left(3 - x\right) \left(- \cot^{2}{\left(\frac{\pi x}{6} \right)} - 1\right) \log{\left(3 - x \right)}^{2}}{6}\right)$$
=
$$\lim_{x \to 3^+}\left(- \frac{\pi \left(3 - x\right) \left(- \frac{\cot^{2}{\left(\frac{\pi x}{6} \right)}}{6} - \frac{1}{6}\right) \log{\left(3 - x \right)}^{2}}{- \cot^{2}{\left(\frac{\pi x}{6} \right)} - 1}\right)$$
=
$$\lim_{x \to 3^+}\left(\frac{\frac{d}{d x} \left(- \pi \left(3 - x\right) \left(- \frac{\cot^{2}{\left(\frac{\pi x}{6} \right)}}{6} - \frac{1}{6}\right) \log{\left(3 - x \right)}^{2}\right)}{\frac{d}{d x} \left(- \cot^{2}{\left(\frac{\pi x}{6} \right)} - 1\right)}\right)$$
=
$$\lim_{x \to 3^+}\left(- \frac{3 \left(\frac{\pi^{2} \left(3 - x\right) \left(- \cot^{2}{\left(\frac{\pi x}{6} \right)} - 1\right) \log{\left(3 - x \right)}^{2} \cot{\left(\frac{\pi x}{6} \right)}}{18} + \pi \left(- \frac{\cot^{2}{\left(\frac{\pi x}{6} \right)}}{6} - \frac{1}{6}\right) \log{\left(3 - x \right)}^{2} + 2 \pi \left(- \frac{\cot^{2}{\left(\frac{\pi x}{6} \right)}}{6} - \frac{1}{6}\right) \log{\left(3 - x \right)}\right)}{\pi \left(- \cot^{2}{\left(\frac{\pi x}{6} \right)} - 1\right) \cot{\left(\frac{\pi x}{6} \right)}}\right)$$
=
$$\lim_{x \to 3^+}\left(\frac{3 \left(\frac{\pi^{2} x \log{\left(3 - x \right)}^{2} \cot^{3}{\left(\frac{\pi x}{6} \right)}}{18} + \frac{\pi^{2} x \log{\left(3 - x \right)}^{2} \cot{\left(\frac{\pi x}{6} \right)}}{18} - \frac{\pi^{2} \log{\left(3 - x \right)}^{2} \cot^{3}{\left(\frac{\pi x}{6} \right)}}{6} - \frac{\pi \log{\left(3 - x \right)}^{2} \cot^{2}{\left(\frac{\pi x}{6} \right)}}{6} - \frac{\pi^{2} \log{\left(3 - x \right)}^{2} \cot{\left(\frac{\pi x}{6} \right)}}{6} - \frac{\pi \log{\left(3 - x \right)}^{2}}{6} - \frac{\pi \log{\left(3 - x \right)} \cot^{2}{\left(\frac{\pi x}{6} \right)}}{3} - \frac{\pi \log{\left(3 - x \right)}}{3}\right)}{\pi \cot{\left(\frac{\pi x}{6} \right)}}\right)$$
=
$$\lim_{x \to 3^+}\left(\frac{3 \left(\frac{\pi^{2} x \log{\left(3 - x \right)}^{2} \cot^{3}{\left(\frac{\pi x}{6} \right)}}{18} + \frac{\pi^{2} x \log{\left(3 - x \right)}^{2} \cot{\left(\frac{\pi x}{6} \right)}}{18} - \frac{\pi^{2} \log{\left(3 - x \right)}^{2} \cot^{3}{\left(\frac{\pi x}{6} \right)}}{6} - \frac{\pi \log{\left(3 - x \right)}^{2} \cot^{2}{\left(\frac{\pi x}{6} \right)}}{6} - \frac{\pi^{2} \log{\left(3 - x \right)}^{2} \cot{\left(\frac{\pi x}{6} \right)}}{6} - \frac{\pi \log{\left(3 - x \right)}^{2}}{6} - \frac{\pi \log{\left(3 - x \right)} \cot^{2}{\left(\frac{\pi x}{6} \right)}}{3} - \frac{\pi \log{\left(3 - x \right)}}{3}\right)}{\pi \cot{\left(\frac{\pi x}{6} \right)}}\right)$$
=
$$0$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 2 vez (veces)
0/0,
tal que el límite para el numerador es
$$\lim_{x \to 3^+} \cot{\left(\frac{\pi x}{6} \right)} = 0$$
y el límite para el denominador es
$$\lim_{x \to 3^+} \frac{1}{\log{\left(3 - x \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 3^+}\left(\log{\left(3 - x \right)} \cot{\left(\frac{\pi x}{6} \right)}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 3^+}\left(\log{\left(3 - x \right)} \cot{\left(\frac{\pi x}{6} \right)}\right)$$
=
$$\lim_{x \to 3^+}\left(\frac{\frac{d}{d x} \cot{\left(\frac{\pi x}{6} \right)}}{\frac{d}{d x} \frac{1}{\log{\left(3 - x \right)}}}\right)$$
=
$$\lim_{x \to 3^+}\left(\frac{\pi \left(3 - x\right) \left(- \cot^{2}{\left(\frac{\pi x}{6} \right)} - 1\right) \log{\left(3 - x \right)}^{2}}{6}\right)$$
=
$$\lim_{x \to 3^+}\left(- \frac{\pi \left(3 - x\right) \left(- \frac{\cot^{2}{\left(\frac{\pi x}{6} \right)}}{6} - \frac{1}{6}\right) \log{\left(3 - x \right)}^{2}}{- \cot^{2}{\left(\frac{\pi x}{6} \right)} - 1}\right)$$
=
$$\lim_{x \to 3^+}\left(\frac{\frac{d}{d x} \left(- \pi \left(3 - x\right) \left(- \frac{\cot^{2}{\left(\frac{\pi x}{6} \right)}}{6} - \frac{1}{6}\right) \log{\left(3 - x \right)}^{2}\right)}{\frac{d}{d x} \left(- \cot^{2}{\left(\frac{\pi x}{6} \right)} - 1\right)}\right)$$
=
$$\lim_{x \to 3^+}\left(- \frac{3 \left(\frac{\pi^{2} \left(3 - x\right) \left(- \cot^{2}{\left(\frac{\pi x}{6} \right)} - 1\right) \log{\left(3 - x \right)}^{2} \cot{\left(\frac{\pi x}{6} \right)}}{18} + \pi \left(- \frac{\cot^{2}{\left(\frac{\pi x}{6} \right)}}{6} - \frac{1}{6}\right) \log{\left(3 - x \right)}^{2} + 2 \pi \left(- \frac{\cot^{2}{\left(\frac{\pi x}{6} \right)}}{6} - \frac{1}{6}\right) \log{\left(3 - x \right)}\right)}{\pi \left(- \cot^{2}{\left(\frac{\pi x}{6} \right)} - 1\right) \cot{\left(\frac{\pi x}{6} \right)}}\right)$$
=
$$\lim_{x \to 3^+}\left(\frac{3 \left(\frac{\pi^{2} x \log{\left(3 - x \right)}^{2} \cot^{3}{\left(\frac{\pi x}{6} \right)}}{18} + \frac{\pi^{2} x \log{\left(3 - x \right)}^{2} \cot{\left(\frac{\pi x}{6} \right)}}{18} - \frac{\pi^{2} \log{\left(3 - x \right)}^{2} \cot^{3}{\left(\frac{\pi x}{6} \right)}}{6} - \frac{\pi \log{\left(3 - x \right)}^{2} \cot^{2}{\left(\frac{\pi x}{6} \right)}}{6} - \frac{\pi^{2} \log{\left(3 - x \right)}^{2} \cot{\left(\frac{\pi x}{6} \right)}}{6} - \frac{\pi \log{\left(3 - x \right)}^{2}}{6} - \frac{\pi \log{\left(3 - x \right)} \cot^{2}{\left(\frac{\pi x}{6} \right)}}{3} - \frac{\pi \log{\left(3 - x \right)}}{3}\right)}{\pi \cot{\left(\frac{\pi x}{6} \right)}}\right)$$
=
$$\lim_{x \to 3^+}\left(\frac{3 \left(\frac{\pi^{2} x \log{\left(3 - x \right)}^{2} \cot^{3}{\left(\frac{\pi x}{6} \right)}}{18} + \frac{\pi^{2} x \log{\left(3 - x \right)}^{2} \cot{\left(\frac{\pi x}{6} \right)}}{18} - \frac{\pi^{2} \log{\left(3 - x \right)}^{2} \cot^{3}{\left(\frac{\pi x}{6} \right)}}{6} - \frac{\pi \log{\left(3 - x \right)}^{2} \cot^{2}{\left(\frac{\pi x}{6} \right)}}{6} - \frac{\pi^{2} \log{\left(3 - x \right)}^{2} \cot{\left(\frac{\pi x}{6} \right)}}{6} - \frac{\pi \log{\left(3 - x \right)}^{2}}{6} - \frac{\pi \log{\left(3 - x \right)} \cot^{2}{\left(\frac{\pi x}{6} \right)}}{3} - \frac{\pi \log{\left(3 - x \right)}}{3}\right)}{\pi \cot{\left(\frac{\pi x}{6} \right)}}\right)$$
=
$$0$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 2 vez (veces)
A la izquierda y a la derecha
[src]
/ /pi*x\ \ lim |cot|----|*log(3 - x)| x->3+\ \ 6 / /
$$\lim_{x \to 3^+}\left(\log{\left(3 - x \right)} \cot{\left(\frac{\pi x}{6} \right)}\right)$$
0
$$0$$
= (0.000976108248528767 - 0.000414013431447878j)
/ /pi*x\ \ lim |cot|----|*log(3 - x)| x->3-\ \ 6 / /
$$\lim_{x \to 3^-}\left(\log{\left(3 - x \right)} \cot{\left(\frac{\pi x}{6} \right)}\right)$$
0
$$0$$
= -0.00102337405546272
= -0.00102337405546272
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 3^-}\left(\log{\left(3 - x \right)} \cot{\left(\frac{\pi x}{6} \right)}\right) = 0$$
Más detalles con x→3 a la izquierda
$$\lim_{x \to 3^+}\left(\log{\left(3 - x \right)} \cot{\left(\frac{\pi x}{6} \right)}\right) = 0$$
$$\lim_{x \to \infty}\left(\log{\left(3 - x \right)} \cot{\left(\frac{\pi x}{6} \right)}\right)$$
Más detalles con x→oo
$$\lim_{x \to 0^-}\left(\log{\left(3 - x \right)} \cot{\left(\frac{\pi x}{6} \right)}\right) = -\infty$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\log{\left(3 - x \right)} \cot{\left(\frac{\pi x}{6} \right)}\right) = \infty$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\log{\left(3 - x \right)} \cot{\left(\frac{\pi x}{6} \right)}\right) = \sqrt{3} \log{\left(2 \right)}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\log{\left(3 - x \right)} \cot{\left(\frac{\pi x}{6} \right)}\right) = \sqrt{3} \log{\left(2 \right)}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\log{\left(3 - x \right)} \cot{\left(\frac{\pi x}{6} \right)}\right)$$
Más detalles con x→-oo
Más detalles con x→3 a la izquierda
$$\lim_{x \to 3^+}\left(\log{\left(3 - x \right)} \cot{\left(\frac{\pi x}{6} \right)}\right) = 0$$
$$\lim_{x \to \infty}\left(\log{\left(3 - x \right)} \cot{\left(\frac{\pi x}{6} \right)}\right)$$
Más detalles con x→oo
$$\lim_{x \to 0^-}\left(\log{\left(3 - x \right)} \cot{\left(\frac{\pi x}{6} \right)}\right) = -\infty$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\log{\left(3 - x \right)} \cot{\left(\frac{\pi x}{6} \right)}\right) = \infty$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\log{\left(3 - x \right)} \cot{\left(\frac{\pi x}{6} \right)}\right) = \sqrt{3} \log{\left(2 \right)}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\log{\left(3 - x \right)} \cot{\left(\frac{\pi x}{6} \right)}\right) = \sqrt{3} \log{\left(2 \right)}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\log{\left(3 - x \right)} \cot{\left(\frac{\pi x}{6} \right)}\right)$$
Más detalles con x→-oo
Respuesta numérica
[src]
(0.000976108248528767 - 0.000414013431447878j)
(0.000976108248528767 - 0.000414013431447878j)