Sr Examen
Otras calculadoras:
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- Límite de la función:
-
Límite de (2*x/(1+2*x))^x
-
Límite de (5+x)/(-6+3*x)
-
Límite de (1-sqrt(1-x^2))/x^2
-
Límite de (-2+x^3-3*x)/(-2+x)
-
Expresiones idénticas
- cot(t)*log(t*cot(t))
- cotangente de (t) multiplicar por logaritmo de (t multiplicar por cotangente de (t))
- cot(t)log(tcot(t))
- cottlogtcott
-
Expresiones con funciones
- Cotangente cot
- cot(7*x)^tan(x)
- cot(9*x)^(1/log(x))
- cot(x)/log(10)
- cot(x)^cot(x)
- cot(x)^2*(1-cos(x))
- Logaritmo log
- log(sqrt((1+x)/(1-x)))/x
- log(x-a)/log(e^x-e^a)
- log(-5+x)/log(e^x-e^5)
- log(5+x)/(3+x)^(1/4)
- log(sin(x))/(-pi+2*x)^2
- Cotangente cot
- cot(7*x)^tan(x)
- cot(9*x)^(1/log(x))
- cot(x)/log(10)
- cot(x)^cot(x)
- cot(x)^2*(1-cos(x))
Límite de la función cot(t)*log(t*cot(t))
Solución
Ha introducido
[src]
lim (cot(t)*log(t*cot(t))) pi t->--+ 2
$$\lim_{t \to \frac{\pi}{2}^+}\left(\log{\left(t \cot{\left(t \right)} \right)} \cot{\left(t \right)}\right)$$
Limit(cot(t)*log(t*cot(t)), t, pi/2)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{t \to \frac{\pi}{2}^+} \cot{\left(t \right)} = 0$$
y el límite para el denominador es
$$\lim_{t \to \frac{\pi}{2}^+} \frac{1}{\log{\left(t \cot{\left(t \right)} \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{t \to \frac{\pi}{2}^+}\left(\log{\left(t \cot{\left(t \right)} \right)} \cot{\left(t \right)}\right)$$
=
$$\lim_{t \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d t} \cot{\left(t \right)}}{\frac{d}{d t} \frac{1}{\log{\left(t \cot{\left(t \right)} \right)}}}\right)$$
=
$$\lim_{t \to \frac{\pi}{2}^+}\left(- \frac{t \left(- \cot^{2}{\left(t \right)} - 1\right) \log{\left(t \cot{\left(t \right)} \right)}^{2} \cot{\left(t \right)}}{t \left(- \cot^{2}{\left(t \right)} - 1\right) + \cot{\left(t \right)}}\right)$$
=
$$\lim_{t \to \frac{\pi}{2}^+}\left(\left(- \cot^{2}{\left(t \right)} - 1\right) \log{\left(t \cot{\left(t \right)} \right)}^{2} \cot{\left(t \right)}\right)$$
=
$$\lim_{t \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d t} \log{\left(t \cot{\left(t \right)} \right)}^{2} \cot{\left(t \right)}}{\frac{d}{d t} \frac{1}{- \cot^{2}{\left(t \right)} - 1}}\right)$$
=
$$\lim_{t \to \frac{\pi}{2}^+}\left(\frac{\left(\left(- \cot^{2}{\left(t \right)} - 1\right) \log{\left(t \cot{\left(t \right)} \right)}^{2} + \frac{2 \left(t \left(- \cot^{2}{\left(t \right)} - 1\right) + \cot{\left(t \right)}\right) \log{\left(t \cot{\left(t \right)} \right)}}{t}\right) \left(- \cot^{2}{\left(t \right)} - 1\right)^{2}}{\left(- 2 \cot^{2}{\left(t \right)} - 2\right) \cot{\left(t \right)}}\right)$$
=
$$\lim_{t \to \frac{\pi}{2}^+}\left(- \frac{- \log{\left(t \cot{\left(t \right)} \right)}^{2} \cot^{2}{\left(t \right)} - \log{\left(t \cot{\left(t \right)} \right)}^{2} - 2 \log{\left(t \cot{\left(t \right)} \right)} \cot^{2}{\left(t \right)} - 2 \log{\left(t \cot{\left(t \right)} \right)} + \frac{2 \log{\left(t \cot{\left(t \right)} \right)} \cot{\left(t \right)}}{t}}{2 \cot{\left(t \right)}}\right)$$
=
$$\lim_{t \to \frac{\pi}{2}^+}\left(- \frac{- \log{\left(t \cot{\left(t \right)} \right)}^{2} \cot^{2}{\left(t \right)} - \log{\left(t \cot{\left(t \right)} \right)}^{2} - 2 \log{\left(t \cot{\left(t \right)} \right)} \cot^{2}{\left(t \right)} - 2 \log{\left(t \cot{\left(t \right)} \right)} + \frac{2 \log{\left(t \cot{\left(t \right)} \right)} \cot{\left(t \right)}}{t}}{2 \cot{\left(t \right)}}\right)$$
=
$$0$$
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
$$\lim_{t \to \frac{\pi}{2}^+} \cot{\left(t \right)} = 0$$
y el límite para el denominador es
$$\lim_{t \to \frac{\pi}{2}^+} \frac{1}{\log{\left(t \cot{\left(t \right)} \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{t \to \frac{\pi}{2}^+}\left(\log{\left(t \cot{\left(t \right)} \right)} \cot{\left(t \right)}\right)$$
=
$$\lim_{t \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d t} \cot{\left(t \right)}}{\frac{d}{d t} \frac{1}{\log{\left(t \cot{\left(t \right)} \right)}}}\right)$$
=
$$\lim_{t \to \frac{\pi}{2}^+}\left(- \frac{t \left(- \cot^{2}{\left(t \right)} - 1\right) \log{\left(t \cot{\left(t \right)} \right)}^{2} \cot{\left(t \right)}}{t \left(- \cot^{2}{\left(t \right)} - 1\right) + \cot{\left(t \right)}}\right)$$
=
$$\lim_{t \to \frac{\pi}{2}^+}\left(\left(- \cot^{2}{\left(t \right)} - 1\right) \log{\left(t \cot{\left(t \right)} \right)}^{2} \cot{\left(t \right)}\right)$$
=
$$\lim_{t \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d t} \log{\left(t \cot{\left(t \right)} \right)}^{2} \cot{\left(t \right)}}{\frac{d}{d t} \frac{1}{- \cot^{2}{\left(t \right)} - 1}}\right)$$
=
$$\lim_{t \to \frac{\pi}{2}^+}\left(\frac{\left(\left(- \cot^{2}{\left(t \right)} - 1\right) \log{\left(t \cot{\left(t \right)} \right)}^{2} + \frac{2 \left(t \left(- \cot^{2}{\left(t \right)} - 1\right) + \cot{\left(t \right)}\right) \log{\left(t \cot{\left(t \right)} \right)}}{t}\right) \left(- \cot^{2}{\left(t \right)} - 1\right)^{2}}{\left(- 2 \cot^{2}{\left(t \right)} - 2\right) \cot{\left(t \right)}}\right)$$
=
$$\lim_{t \to \frac{\pi}{2}^+}\left(- \frac{- \log{\left(t \cot{\left(t \right)} \right)}^{2} \cot^{2}{\left(t \right)} - \log{\left(t \cot{\left(t \right)} \right)}^{2} - 2 \log{\left(t \cot{\left(t \right)} \right)} \cot^{2}{\left(t \right)} - 2 \log{\left(t \cot{\left(t \right)} \right)} + \frac{2 \log{\left(t \cot{\left(t \right)} \right)} \cot{\left(t \right)}}{t}}{2 \cot{\left(t \right)}}\right)$$
=
$$\lim_{t \to \frac{\pi}{2}^+}\left(- \frac{- \log{\left(t \cot{\left(t \right)} \right)}^{2} \cot^{2}{\left(t \right)} - \log{\left(t \cot{\left(t \right)} \right)}^{2} - 2 \log{\left(t \cot{\left(t \right)} \right)} \cot^{2}{\left(t \right)} - 2 \log{\left(t \cot{\left(t \right)} \right)} + \frac{2 \log{\left(t \cot{\left(t \right)} \right)} \cot{\left(t \right)}}{t}}{2 \cot{\left(t \right)}}\right)$$
=
$$0$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 2 vez (veces)
Gráfica
A la izquierda y a la derecha
[src]
lim (cot(t)*log(t*cot(t))) pi t->--+ 2
$$\lim_{t \to \frac{\pi}{2}^+}\left(\log{\left(t \cot{\left(t \right)} \right)} \cot{\left(t \right)}\right)$$
0
$$0$$
= (0.00183324192022206 - 0.000852522131236593j)
lim (cot(t)*log(t*cot(t))) pi t->--- 2
$$\lim_{t \to \frac{\pi}{2}^-}\left(\log{\left(t \cot{\left(t \right)} \right)} \cot{\left(t \right)}\right)$$
0
$$0$$
= -0.00191755035984963
= -0.00191755035984963
Otros límites con t→0, -oo, +oo, 1
$$\lim_{t \to \frac{\pi}{2}^-}\left(\log{\left(t \cot{\left(t \right)} \right)} \cot{\left(t \right)}\right) = 0$$
Más detalles con t→pi/2 a la izquierda
$$\lim_{t \to \frac{\pi}{2}^+}\left(\log{\left(t \cot{\left(t \right)} \right)} \cot{\left(t \right)}\right) = 0$$
$$\lim_{t \to \infty}\left(\log{\left(t \cot{\left(t \right)} \right)} \cot{\left(t \right)}\right)$$
Más detalles con t→oo
$$\lim_{t \to 0^-}\left(\log{\left(t \cot{\left(t \right)} \right)} \cot{\left(t \right)}\right) = 0$$
Más detalles con t→0 a la izquierda
$$\lim_{t \to 0^+}\left(\log{\left(t \cot{\left(t \right)} \right)} \cot{\left(t \right)}\right) = 0$$
Más detalles con t→0 a la derecha
$$\lim_{t \to 1^-}\left(\log{\left(t \cot{\left(t \right)} \right)} \cot{\left(t \right)}\right) = - \frac{\log{\left(\tan{\left(1 \right)} \right)}}{\tan{\left(1 \right)}}$$
Más detalles con t→1 a la izquierda
$$\lim_{t \to 1^+}\left(\log{\left(t \cot{\left(t \right)} \right)} \cot{\left(t \right)}\right) = - \frac{\log{\left(\tan{\left(1 \right)} \right)}}{\tan{\left(1 \right)}}$$
Más detalles con t→1 a la derecha
$$\lim_{t \to -\infty}\left(\log{\left(t \cot{\left(t \right)} \right)} \cot{\left(t \right)}\right)$$
Más detalles con t→-oo
Más detalles con t→pi/2 a la izquierda
$$\lim_{t \to \frac{\pi}{2}^+}\left(\log{\left(t \cot{\left(t \right)} \right)} \cot{\left(t \right)}\right) = 0$$
$$\lim_{t \to \infty}\left(\log{\left(t \cot{\left(t \right)} \right)} \cot{\left(t \right)}\right)$$
Más detalles con t→oo
$$\lim_{t \to 0^-}\left(\log{\left(t \cot{\left(t \right)} \right)} \cot{\left(t \right)}\right) = 0$$
Más detalles con t→0 a la izquierda
$$\lim_{t \to 0^+}\left(\log{\left(t \cot{\left(t \right)} \right)} \cot{\left(t \right)}\right) = 0$$
Más detalles con t→0 a la derecha
$$\lim_{t \to 1^-}\left(\log{\left(t \cot{\left(t \right)} \right)} \cot{\left(t \right)}\right) = - \frac{\log{\left(\tan{\left(1 \right)} \right)}}{\tan{\left(1 \right)}}$$
Más detalles con t→1 a la izquierda
$$\lim_{t \to 1^+}\left(\log{\left(t \cot{\left(t \right)} \right)} \cot{\left(t \right)}\right) = - \frac{\log{\left(\tan{\left(1 \right)} \right)}}{\tan{\left(1 \right)}}$$
Más detalles con t→1 a la derecha
$$\lim_{t \to -\infty}\left(\log{\left(t \cot{\left(t \right)} \right)} \cot{\left(t \right)}\right)$$
Más detalles con t→-oo
Respuesta numérica
[src]
(0.00183324192022206 - 0.000852522131236593j)
(0.00183324192022206 - 0.000852522131236593j)