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- Límite de la función:
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Límite de 4-3*x+2*x^2
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Límite de ((3+x)/(-2+x))^x
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Límite de (-8+x^3)/(-6+x+x^2)
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Límite de (-3+sqrt(1+2*x))/(sqrt(-2+x)-sqrt(2))
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Expresiones idénticas
- asin(dos *x)*cot(tres *x)
- ar coseno de eno de (2 multiplicar por x) multiplicar por cotangente de (3 multiplicar por x)
- ar coseno de eno de (dos multiplicar por x) multiplicar por cotangente de (tres multiplicar por x)
- asin(2x)cot(3x)
- asin2xcot3x
-
Expresiones semejantes
- arcsin(2*x)*cot(3*x)
-
Expresiones con funciones
- Arcoseno arcsin
- asin(1/x)
- asin(8*x)/log(1-4*x)
- asin(4*x^2)/(2*x)
- asin(6*x)^2/(x*tan(3*x))
- asin(1/x)*log(x)
- Cotangente cot
- cot(3*x)/cot(2*x)
- cot(2*x)/log(x)
- cot(4*x)*tan(6*x)
- cot(x)^2/(2-p-x)^4
- cot(x)/log(-1+e^x)
Límite de la función asin(2*x)*cot(3*x)
Solución
Ha introducido
[src]
lim (asin(2*x)*cot(3*x)) x->0+
$$\lim_{x \to 0^+}\left(\cot{\left(3 x \right)} \operatorname{asin}{\left(2 x \right)}\right)$$
Limit(asin(2*x)*cot(3*x), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+} \operatorname{asin}{\left(2 x \right)} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(3 x \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\cot{\left(3 x \right)} \operatorname{asin}{\left(2 x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \operatorname{asin}{\left(2 x \right)}}{\frac{d}{d x} \frac{1}{\cot{\left(3 x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \cot^{2}{\left(3 x \right)}}{\sqrt{1 - 4 x^{2}} \left(3 \cot^{2}{\left(3 x \right)} + 3\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \cot^{2}{\left(3 x \right)}}{3 \cot^{2}{\left(3 x \right)} + 3}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{3 \cot^{2}{\left(3 x \right)} + 3}}{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(3 x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\left(6 \cot^{2}{\left(3 x \right)} + 6\right) \left(- \frac{9 \cot^{4}{\left(3 x \right)}}{- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}} - \frac{18 \cot^{2}{\left(3 x \right)}}{- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}} - \frac{9}{- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{6 \cot^{2}{\left(3 x \right)} + 6}}{\frac{d}{d x} \left(- \frac{9 \cot^{4}{\left(3 x \right)}}{- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}} - \frac{18 \cot^{2}{\left(3 x \right)}}{- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}} - \frac{9}{- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{6 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)}}{\left(6 \cot^{2}{\left(3 x \right)} + 6\right)^{2} \left(- \frac{9 \left(36 \left(- 18 \cot^{2}{\left(3 x \right)} - 18\right) \cot^{5}{\left(3 x \right)} + 36 \left(- 12 \cot^{2}{\left(3 x \right)} - 12\right) \cot^{3}{\left(3 x \right)}\right) \cot^{4}{\left(3 x \right)}}{\left(- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}\right)^{2}} - \frac{18 \left(36 \left(- 18 \cot^{2}{\left(3 x \right)} - 18\right) \cot^{5}{\left(3 x \right)} + 36 \left(- 12 \cot^{2}{\left(3 x \right)} - 12\right) \cot^{3}{\left(3 x \right)}\right) \cot^{2}{\left(3 x \right)}}{\left(- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}\right)^{2}} - \frac{9 \left(36 \left(- 18 \cot^{2}{\left(3 x \right)} - 18\right) \cot^{5}{\left(3 x \right)} + 36 \left(- 12 \cot^{2}{\left(3 x \right)} - 12\right) \cot^{3}{\left(3 x \right)}\right)}{\left(- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}\right)^{2}} - \frac{9 \left(- 12 \cot^{2}{\left(3 x \right)} - 12\right) \cot^{3}{\left(3 x \right)}}{- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}} - \frac{18 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)}}{- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{6 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)}}{\left(6 \cot^{2}{\left(3 x \right)} + 6\right)^{2} \left(- \frac{9 \left(36 \left(- 18 \cot^{2}{\left(3 x \right)} - 18\right) \cot^{5}{\left(3 x \right)} + 36 \left(- 12 \cot^{2}{\left(3 x \right)} - 12\right) \cot^{3}{\left(3 x \right)}\right) \cot^{4}{\left(3 x \right)}}{\left(- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}\right)^{2}} - \frac{18 \left(36 \left(- 18 \cot^{2}{\left(3 x \right)} - 18\right) \cot^{5}{\left(3 x \right)} + 36 \left(- 12 \cot^{2}{\left(3 x \right)} - 12\right) \cot^{3}{\left(3 x \right)}\right) \cot^{2}{\left(3 x \right)}}{\left(- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}\right)^{2}} - \frac{9 \left(36 \left(- 18 \cot^{2}{\left(3 x \right)} - 18\right) \cot^{5}{\left(3 x \right)} + 36 \left(- 12 \cot^{2}{\left(3 x \right)} - 12\right) \cot^{3}{\left(3 x \right)}\right)}{\left(- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}\right)^{2}} - \frac{9 \left(- 12 \cot^{2}{\left(3 x \right)} - 12\right) \cot^{3}{\left(3 x \right)}}{- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}} - \frac{18 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)}}{- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}}\right)}\right)$$
=
$$\frac{2}{3}$$
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
$$\lim_{x \to 0^+} \operatorname{asin}{\left(2 x \right)} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(3 x \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\cot{\left(3 x \right)} \operatorname{asin}{\left(2 x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \operatorname{asin}{\left(2 x \right)}}{\frac{d}{d x} \frac{1}{\cot{\left(3 x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \cot^{2}{\left(3 x \right)}}{\sqrt{1 - 4 x^{2}} \left(3 \cot^{2}{\left(3 x \right)} + 3\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \cot^{2}{\left(3 x \right)}}{3 \cot^{2}{\left(3 x \right)} + 3}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{3 \cot^{2}{\left(3 x \right)} + 3}}{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(3 x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\left(6 \cot^{2}{\left(3 x \right)} + 6\right) \left(- \frac{9 \cot^{4}{\left(3 x \right)}}{- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}} - \frac{18 \cot^{2}{\left(3 x \right)}}{- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}} - \frac{9}{- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{6 \cot^{2}{\left(3 x \right)} + 6}}{\frac{d}{d x} \left(- \frac{9 \cot^{4}{\left(3 x \right)}}{- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}} - \frac{18 \cot^{2}{\left(3 x \right)}}{- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}} - \frac{9}{- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{6 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)}}{\left(6 \cot^{2}{\left(3 x \right)} + 6\right)^{2} \left(- \frac{9 \left(36 \left(- 18 \cot^{2}{\left(3 x \right)} - 18\right) \cot^{5}{\left(3 x \right)} + 36 \left(- 12 \cot^{2}{\left(3 x \right)} - 12\right) \cot^{3}{\left(3 x \right)}\right) \cot^{4}{\left(3 x \right)}}{\left(- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}\right)^{2}} - \frac{18 \left(36 \left(- 18 \cot^{2}{\left(3 x \right)} - 18\right) \cot^{5}{\left(3 x \right)} + 36 \left(- 12 \cot^{2}{\left(3 x \right)} - 12\right) \cot^{3}{\left(3 x \right)}\right) \cot^{2}{\left(3 x \right)}}{\left(- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}\right)^{2}} - \frac{9 \left(36 \left(- 18 \cot^{2}{\left(3 x \right)} - 18\right) \cot^{5}{\left(3 x \right)} + 36 \left(- 12 \cot^{2}{\left(3 x \right)} - 12\right) \cot^{3}{\left(3 x \right)}\right)}{\left(- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}\right)^{2}} - \frac{9 \left(- 12 \cot^{2}{\left(3 x \right)} - 12\right) \cot^{3}{\left(3 x \right)}}{- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}} - \frac{18 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)}}{- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{6 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)}}{\left(6 \cot^{2}{\left(3 x \right)} + 6\right)^{2} \left(- \frac{9 \left(36 \left(- 18 \cot^{2}{\left(3 x \right)} - 18\right) \cot^{5}{\left(3 x \right)} + 36 \left(- 12 \cot^{2}{\left(3 x \right)} - 12\right) \cot^{3}{\left(3 x \right)}\right) \cot^{4}{\left(3 x \right)}}{\left(- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}\right)^{2}} - \frac{18 \left(36 \left(- 18 \cot^{2}{\left(3 x \right)} - 18\right) \cot^{5}{\left(3 x \right)} + 36 \left(- 12 \cot^{2}{\left(3 x \right)} - 12\right) \cot^{3}{\left(3 x \right)}\right) \cot^{2}{\left(3 x \right)}}{\left(- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}\right)^{2}} - \frac{9 \left(36 \left(- 18 \cot^{2}{\left(3 x \right)} - 18\right) \cot^{5}{\left(3 x \right)} + 36 \left(- 12 \cot^{2}{\left(3 x \right)} - 12\right) \cot^{3}{\left(3 x \right)}\right)}{\left(- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}\right)^{2}} - \frac{9 \left(- 12 \cot^{2}{\left(3 x \right)} - 12\right) \cot^{3}{\left(3 x \right)}}{- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}} - \frac{18 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)}}{- 36 \cot^{6}{\left(3 x \right)} - 36 \cot^{4}{\left(3 x \right)}}\right)}\right)$$
=
$$\frac{2}{3}$$
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
$$\lim_{x \to 0^-}\left(\cot{\left(3 x \right)} \operatorname{asin}{\left(2 x \right)}\right) = \frac{2}{3}$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\cot{\left(3 x \right)} \operatorname{asin}{\left(2 x \right)}\right) = \frac{2}{3}$$
$$\lim_{x \to \infty}\left(\cot{\left(3 x \right)} \operatorname{asin}{\left(2 x \right)}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\cot{\left(3 x \right)} \operatorname{asin}{\left(2 x \right)}\right) = \frac{\operatorname{asin}{\left(2 \right)}}{\tan{\left(3 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\cot{\left(3 x \right)} \operatorname{asin}{\left(2 x \right)}\right) = \frac{\operatorname{asin}{\left(2 \right)}}{\tan{\left(3 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\cot{\left(3 x \right)} \operatorname{asin}{\left(2 x \right)}\right)$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\cot{\left(3 x \right)} \operatorname{asin}{\left(2 x \right)}\right) = \frac{2}{3}$$
$$\lim_{x \to \infty}\left(\cot{\left(3 x \right)} \operatorname{asin}{\left(2 x \right)}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\cot{\left(3 x \right)} \operatorname{asin}{\left(2 x \right)}\right) = \frac{\operatorname{asin}{\left(2 \right)}}{\tan{\left(3 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\cot{\left(3 x \right)} \operatorname{asin}{\left(2 x \right)}\right) = \frac{\operatorname{asin}{\left(2 \right)}}{\tan{\left(3 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\cot{\left(3 x \right)} \operatorname{asin}{\left(2 x \right)}\right)$$
Más detalles con x→-oo
A la izquierda y a la derecha
[src]
lim (asin(2*x)*cot(3*x)) x->0+
$$\lim_{x \to 0^+}\left(\cot{\left(3 x \right)} \operatorname{asin}{\left(2 x \right)}\right)$$
2/3
$$\frac{2}{3}$$
= 0.666666666666667
lim (asin(2*x)*cot(3*x)) x->0-
$$\lim_{x \to 0^-}\left(\cot{\left(3 x \right)} \operatorname{asin}{\left(2 x \right)}\right)$$
2/3
$$\frac{2}{3}$$
= 0.666666666666667
= 0.666666666666667