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Otras calculadoras:
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- Límite de la función:
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Límite de (-10-x+3*x^2)/(-10-x^2+7*x)
-
Límite de (-1+sqrt(1+x))/x
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Límite de sin(2*x)/sin(3*x)
-
Límite de sin(5*x)/(2*x)
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Expresiones idénticas
- cot(x)*sin(x/ tres)
- cotangente de (x) multiplicar por seno de (x dividir por 3)
- cotangente de (x) multiplicar por seno de (x dividir por tres)
- cot(x)sin(x/3)
- cotxsinx/3
- cot(x)*sin(x dividir por 3)
-
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)
- Seno sin
- sin(2*x)/sin(3*x)
- sin(5*x)/(2*x)
- sin(3*x)/tan(2*x)
- sin(x)^2/x^2
- sin(a*x)/sin(b*x)
Límite de la función cot(x)*sin(x/3)
Solución
Ha introducido
[src]
/ /x\\ lim |cot(x)*sin|-|| x->0+\ \3//
$$\lim_{x \to 0^+}\left(\sin{\left(\frac{x}{3} \right)} \cot{\left(x \right)}\right)$$
Limit(cot(x)*sin(x/3), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+} \sin{\left(\frac{x}{3} \right)} = 0$$
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.
$$\lim_{x \to 0^+}\left(\sin{\left(\frac{x}{3} \right)} \cot{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(\frac{x}{3} \right)}}{\frac{d}{d x} \frac{1}{\cot{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(\frac{x}{3} \right)} \cot^{2}{\left(x \right)}}{3 \left(\cot^{2}{\left(x \right)} + 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cot^{2}{\left(x \right)}}{3 \left(\cot^{2}{\left(x \right)} + 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot^{2}{\left(x \right)} + 1}}{\frac{d}{d x} \frac{3}{\cot^{2}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\left(2 \cot^{2}{\left(x \right)} + 2\right) \left(- \frac{3 \cot^{4}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{6 \cot^{2}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{3}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(x \right)} + 2}}{\frac{d}{d x} \left(- \frac{3 \cot^{4}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{6 \cot^{2}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{3}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{3 \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{6 \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{3 \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right)}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{3 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{6 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{3 \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{6 \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{3 \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right)}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{3 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{6 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\frac{1}{3}$$
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
$$\lim_{x \to 0^+} \sin{\left(\frac{x}{3} \right)} = 0$$
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.
$$\lim_{x \to 0^+}\left(\sin{\left(\frac{x}{3} \right)} \cot{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(\frac{x}{3} \right)}}{\frac{d}{d x} \frac{1}{\cot{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(\frac{x}{3} \right)} \cot^{2}{\left(x \right)}}{3 \left(\cot^{2}{\left(x \right)} + 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cot^{2}{\left(x \right)}}{3 \left(\cot^{2}{\left(x \right)} + 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot^{2}{\left(x \right)} + 1}}{\frac{d}{d x} \frac{3}{\cot^{2}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\left(2 \cot^{2}{\left(x \right)} + 2\right) \left(- \frac{3 \cot^{4}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{6 \cot^{2}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{3}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(x \right)} + 2}}{\frac{d}{d x} \left(- \frac{3 \cot^{4}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{6 \cot^{2}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{3}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{3 \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{6 \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{3 \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right)}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{3 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{6 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{3 \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{6 \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{3 \left(2 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 2 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right)}{\left(- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{3 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}} - \frac{6 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 2 \cot^{6}{\left(x \right)} - 2 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\frac{1}{3}$$
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
$$\lim_{x \to 0^-}\left(\sin{\left(\frac{x}{3} \right)} \cot{\left(x \right)}\right) = \frac{1}{3}$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\sin{\left(\frac{x}{3} \right)} \cot{\left(x \right)}\right) = \frac{1}{3}$$
$$\lim_{x \to \infty}\left(\sin{\left(\frac{x}{3} \right)} \cot{\left(x \right)}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\sin{\left(\frac{x}{3} \right)} \cot{\left(x \right)}\right) = \frac{\sin{\left(\frac{1}{3} \right)}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\sin{\left(\frac{x}{3} \right)} \cot{\left(x \right)}\right) = \frac{\sin{\left(\frac{1}{3} \right)}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\sin{\left(\frac{x}{3} \right)} \cot{\left(x \right)}\right)$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\sin{\left(\frac{x}{3} \right)} \cot{\left(x \right)}\right) = \frac{1}{3}$$
$$\lim_{x \to \infty}\left(\sin{\left(\frac{x}{3} \right)} \cot{\left(x \right)}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\sin{\left(\frac{x}{3} \right)} \cot{\left(x \right)}\right) = \frac{\sin{\left(\frac{1}{3} \right)}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\sin{\left(\frac{x}{3} \right)} \cot{\left(x \right)}\right) = \frac{\sin{\left(\frac{1}{3} \right)}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\sin{\left(\frac{x}{3} \right)} \cot{\left(x \right)}\right)$$
Más detalles con x→-oo
A la izquierda y a la derecha
[src]
/ /x\\ lim |cot(x)*sin|-|| x->0+\ \3//
$$\lim_{x \to 0^+}\left(\sin{\left(\frac{x}{3} \right)} \cot{\left(x \right)}\right)$$
1/3
$$\frac{1}{3}$$
= 0.333333333333333
/ /x\\ lim |cot(x)*sin|-|| x->0-\ \3//
$$\lim_{x \to 0^-}\left(\sin{\left(\frac{x}{3} \right)} \cot{\left(x \right)}\right)$$
1/3
$$\frac{1}{3}$$
= 0.333333333333333
= 0.333333333333333