Sr Examen
Otras calculadoras:
-
¿Cómo usar?
- Límite de la función:
-
Límite de (1-e^(2*x))*cot(x)
-
Límite de 1/cos(x)
-
Límite de 4+x^2
-
Límite de tan(2*x)/(2*x)
-
Expresiones idénticas
- log(cot(x))/log(x)
- logaritmo de ( cotangente de (x)) dividir por logaritmo de (x)
- logcotx/logx
- log(cot(x)) dividir por log(x)
-
Expresiones con funciones
- Logaritmo log
- log(1+x)/x^2
- log(4-x^2)
- log(x)/x
- log(x/2)/sin(-2+x)^2
- log(2*x+3/sqrt(x))/log(x)
- Cotangente cot
- cot(2*x)*tan(x)
- cot(x)^2/(-pi+2*x)^4
- cot(3*x)*sin(2*x)/cos(4*x)
- cot(3*x)
- cot(x+pi/4)*tan(2*x)
- Logaritmo log
- log(1+x)/x^2
- log(4-x^2)
- log(x)/x
- log(x/2)/sin(-2+x)^2
- log(2*x+3/sqrt(x))/log(x)
Límite de la función log(cot(x))/log(x)
Solución
Ha introducido
[src]
/log(cot(x))\ lim |-----------| x->0+\ log(x) /
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}}{\log{\left(x \right)}}\right)$$
Limit(log(cot(x))/log(x), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+} \frac{1}{\log{\left(x \right)}} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\log{\left(\cot{\left(x \right)} \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}}{\log{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(x \right)}}}{\frac{d}{d x} \frac{1}{\log{\left(\cot{\left(x \right)} \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}^{2} \cot{\left(x \right)}}{x \left(- \cot^{2}{\left(x \right)} - 1\right) \log{\left(x \right)}^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{- \cot^{2}{\left(x \right)} - 1}}{\frac{d}{d x} \frac{x \log{\left(x \right)}^{2}}{\log{\left(\cot{\left(x \right)} \right)}^{2} \cot{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(- \cot^{2}{\left(x \right)} - 1\right)^{2} \left(- \frac{2 x \left(- \cot^{2}{\left(x \right)} - 1\right) \log{\left(x \right)}^{2}}{\log{\left(\cot{\left(x \right)} \right)}^{3} \cot^{2}{\left(x \right)}} + \frac{x \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x \right)}^{2}}{\log{\left(\cot{\left(x \right)} \right)}^{2} \cot^{2}{\left(x \right)}} + \frac{\log{\left(x \right)}^{2}}{\log{\left(\cot{\left(x \right)} \right)}^{2} \cot{\left(x \right)}} + \frac{2 \log{\left(x \right)}}{\log{\left(\cot{\left(x \right)} \right)}^{2} \cot{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(- \cot^{2}{\left(x \right)} - 1\right)^{2} \left(- \frac{2 x \left(- \cot^{2}{\left(x \right)} - 1\right) \log{\left(x \right)}^{2}}{\log{\left(\cot{\left(x \right)} \right)}^{3} \cot^{2}{\left(x \right)}} + \frac{x \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x \right)}^{2}}{\log{\left(\cot{\left(x \right)} \right)}^{2} \cot^{2}{\left(x \right)}} + \frac{\log{\left(x \right)}^{2}}{\log{\left(\cot{\left(x \right)} \right)}^{2} \cot{\left(x \right)}} + \frac{2 \log{\left(x \right)}}{\log{\left(\cot{\left(x \right)} \right)}^{2} \cot{\left(x \right)}}\right)}\right)$$
=
$$-1$$
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
$$\lim_{x \to 0^+} \frac{1}{\log{\left(x \right)}} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\log{\left(\cot{\left(x \right)} \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}}{\log{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(x \right)}}}{\frac{d}{d x} \frac{1}{\log{\left(\cot{\left(x \right)} \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}^{2} \cot{\left(x \right)}}{x \left(- \cot^{2}{\left(x \right)} - 1\right) \log{\left(x \right)}^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{- \cot^{2}{\left(x \right)} - 1}}{\frac{d}{d x} \frac{x \log{\left(x \right)}^{2}}{\log{\left(\cot{\left(x \right)} \right)}^{2} \cot{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(- \cot^{2}{\left(x \right)} - 1\right)^{2} \left(- \frac{2 x \left(- \cot^{2}{\left(x \right)} - 1\right) \log{\left(x \right)}^{2}}{\log{\left(\cot{\left(x \right)} \right)}^{3} \cot^{2}{\left(x \right)}} + \frac{x \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x \right)}^{2}}{\log{\left(\cot{\left(x \right)} \right)}^{2} \cot^{2}{\left(x \right)}} + \frac{\log{\left(x \right)}^{2}}{\log{\left(\cot{\left(x \right)} \right)}^{2} \cot{\left(x \right)}} + \frac{2 \log{\left(x \right)}}{\log{\left(\cot{\left(x \right)} \right)}^{2} \cot{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(- \cot^{2}{\left(x \right)} - 1\right)^{2} \left(- \frac{2 x \left(- \cot^{2}{\left(x \right)} - 1\right) \log{\left(x \right)}^{2}}{\log{\left(\cot{\left(x \right)} \right)}^{3} \cot^{2}{\left(x \right)}} + \frac{x \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x \right)}^{2}}{\log{\left(\cot{\left(x \right)} \right)}^{2} \cot^{2}{\left(x \right)}} + \frac{\log{\left(x \right)}^{2}}{\log{\left(\cot{\left(x \right)} \right)}^{2} \cot{\left(x \right)}} + \frac{2 \log{\left(x \right)}}{\log{\left(\cot{\left(x \right)} \right)}^{2} \cot{\left(x \right)}}\right)}\right)$$
=
$$-1$$
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]
/log(cot(x))\ lim |-----------| x->0+\ log(x) /
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}}{\log{\left(x \right)}}\right)$$
-1
$$-1$$
= -0.999999994245146
/log(cot(x))\ lim |-----------| x->0-\ log(x) /
$$\lim_{x \to 0^-}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}}{\log{\left(x \right)}}\right)$$
-1
$$-1$$
= (-0.764159250732235 - 0.634316847654315j)
= (-0.764159250732235 - 0.634316847654315j)
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}}{\log{\left(x \right)}}\right) = -1$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}}{\log{\left(x \right)}}\right) = -1$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}}{\log{\left(x \right)}}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}}{\log{\left(x \right)}}\right) = \infty$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}}{\log{\left(x \right)}}\right) = -\infty$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}}{\log{\left(x \right)}}\right)$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}}{\log{\left(x \right)}}\right) = -1$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}}{\log{\left(x \right)}}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}}{\log{\left(x \right)}}\right) = \infty$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}}{\log{\left(x \right)}}\right) = -\infty$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}}{\log{\left(x \right)}}\right)$$
Más detalles con x→-oo
Gráfico