Sr Examen
Otras calculadoras:
-
¿Cómo usar?
- Límite de la función:
-
Límite de (-10-x+3*x^2)/(-10-x^2+7*x)
-
Límite de (-1+sqrt(1+x))/x
-
Límite de sin(2*x)/sin(3*x)
-
Límite de sin(5*x)/(2*x)
-
Expresiones idénticas
- cot(x)*log(log(e+x))
- cotangente de (x) multiplicar por logaritmo de ( logaritmo de (e más x))
- cot(x)log(log(e+x))
- cotxlogloge+x
-
Expresiones semejantes
- cot(x)*log(log(e-x))
-
Expresiones con funciones
- Cotangente cot
- cot(4*x)*tan(2*x)
- cot(x/8)^2*tan(5*x/4)^2
- cot(7*x)*tan(5*x)
- cot(pi*x/4)^tan(pi*x/2)
- cot(2*x)/tan(4*x)
- Logaritmo log
- log(1+x)/log(x)
- log(1+sin(x))/sin(4*x)
- log(tan(x))/cos(2*x)
- log(x)/(1-x^3)
- log(cos(2*x))/x^2
- Logaritmo log
- log(1+x)/log(x)
- log(1+sin(x))/sin(4*x)
- log(tan(x))/cos(2*x)
- log(x)/(1-x^3)
- log(cos(2*x))/x^2
Límite de la función cot(x)*log(log(e+x))
Solución
Ha introducido
[src]
lim (cot(x)*log(log(E + x))) x->0+
$$\lim_{x \to 0^+}\left(\log{\left(\log{\left(x + e \right)} \right)} \cot{\left(x \right)}\right)$$
Limit(cot(x)*log(log(E + x)), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+} \log{\left(\log{\left(x + e \right)} \right)} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(x \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\log{\left(\log{\left(x + e \right)} \right)} \cot{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \log{\left(\log{\left(x + e \right)} \right)}}{\frac{d}{d x} \frac{1}{\cot{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cot^{2}{\left(x \right)}}{\left(x + e\right) \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x + e \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cot^{2}{\left(x \right)}}{e \left(\cot^{2}{\left(x \right)} + 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot^{2}{\left(x \right)} + 1}}{\frac{d}{d x} \frac{e}{\cot^{2}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\left(2 \cot^{2}{\left(x \right)} + 2\right) \left(- \frac{e \cot^{4}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{2 e \cot^{2}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{e}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(x \right)} + 2}}{\frac{d}{d x} \left(- \frac{e \cot^{4}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{2 e \cot^{2}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{e}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{e \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 e \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{e \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right)}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{e \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{2 e \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{e \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 e \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{e \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right)}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{e \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{2 e \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$e^{-1}$$
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
$$\lim_{x \to 0^+} \log{\left(\log{\left(x + e \right)} \right)} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(x \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\log{\left(\log{\left(x + e \right)} \right)} \cot{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \log{\left(\log{\left(x + e \right)} \right)}}{\frac{d}{d x} \frac{1}{\cot{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cot^{2}{\left(x \right)}}{\left(x + e\right) \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x + e \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cot^{2}{\left(x \right)}}{e \left(\cot^{2}{\left(x \right)} + 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot^{2}{\left(x \right)} + 1}}{\frac{d}{d x} \frac{e}{\cot^{2}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\left(2 \cot^{2}{\left(x \right)} + 2\right) \left(- \frac{e \cot^{4}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{2 e \cot^{2}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{e}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(x \right)} + 2}}{\frac{d}{d x} \left(- \frac{e \cot^{4}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{2 e \cot^{2}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{e}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{e \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 e \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{e \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right)}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{e \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{2 e \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{e \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 e \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{e \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right)}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{e \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{2 e \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$e^{-1}$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)
Gráfica
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\log{\left(\log{\left(x + e \right)} \right)} \cot{\left(x \right)}\right) = e^{-1}$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\log{\left(\log{\left(x + e \right)} \right)} \cot{\left(x \right)}\right) = e^{-1}$$
$$\lim_{x \to \infty}\left(\log{\left(\log{\left(x + e \right)} \right)} \cot{\left(x \right)}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\log{\left(\log{\left(x + e \right)} \right)} \cot{\left(x \right)}\right) = \frac{\log{\left(\log{\left(1 + e \right)} \right)}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\log{\left(\log{\left(x + e \right)} \right)} \cot{\left(x \right)}\right) = \frac{\log{\left(\log{\left(1 + e \right)} \right)}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\log{\left(\log{\left(x + e \right)} \right)} \cot{\left(x \right)}\right)$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\log{\left(\log{\left(x + e \right)} \right)} \cot{\left(x \right)}\right) = e^{-1}$$
$$\lim_{x \to \infty}\left(\log{\left(\log{\left(x + e \right)} \right)} \cot{\left(x \right)}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\log{\left(\log{\left(x + e \right)} \right)} \cot{\left(x \right)}\right) = \frac{\log{\left(\log{\left(1 + e \right)} \right)}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\log{\left(\log{\left(x + e \right)} \right)} \cot{\left(x \right)}\right) = \frac{\log{\left(\log{\left(1 + e \right)} \right)}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\log{\left(\log{\left(x + e \right)} \right)} \cot{\left(x \right)}\right)$$
Más detalles con x→-oo
A la izquierda y a la derecha
[src]
lim (cot(x)*log(log(E + x))) x->0+
$$\lim_{x \to 0^+}\left(\log{\left(\log{\left(x + e \right)} \right)} \cot{\left(x \right)}\right)$$
-1 e
$$e^{-1}$$
= 0.367879441171442
lim (cot(x)*log(log(E + x))) x->0-
$$\lim_{x \to 0^-}\left(\log{\left(\log{\left(x + e \right)} \right)} \cot{\left(x \right)}\right)$$
-1 e
$$e^{-1}$$
= 0.367879441171442
= 0.367879441171442