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Otras calculadoras:
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- Límite de la función:
-
Límite de (-9+x^2)/(3+x)
-
Límite de x^2/(1-cos(6*x))
-
Límite de (4+x^2-5*x)/(8+x^2-6*x)
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Límite de (-3+x^2-2*x)/(-9+x^2)
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Expresiones idénticas
- cot(tres *x)/cot(cinco *x)
- cotangente de (3 multiplicar por x) dividir por cotangente de (5 multiplicar por x)
- cotangente de (tres multiplicar por x) dividir por cotangente de (cinco multiplicar por x)
- cot(3x)/cot(5x)
- cot3x/cot5x
- cot(3*x) dividir por cot(5*x)
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Expresiones con funciones
- Cotangente cot
- cot(2*x)*sin(5*x)
- cot(3*x)*sin(6*x)
- cot(6*x)*sin(3*x)
- cot(-3+x)
- cot(x)*tan(x)
- Cotangente cot
- cot(2*x)*sin(5*x)
- cot(3*x)*sin(6*x)
- cot(6*x)*sin(3*x)
- cot(-3+x)
- cot(x)*tan(x)
Límite de la función cot(3*x)/cot(5*x)
Solución
Ha introducido
[src]
/cot(3*x)\ lim |--------| x->0+\cot(5*x)/
$$\lim_{x \to 0^+}\left(\frac{\cot{\left(3 x \right)}}{\cot{\left(5 x \right)}}\right)$$
Limit(cot(3*x)/cot(5*x), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(5 x \right)}} = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{\cot{\left(3 x \right)}}{\cot{\left(5 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot{\left(5 x \right)}}}{\frac{d}{d x} \frac{1}{\cot{\left(3 x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5 \cot^{2}{\left(3 x \right)} \cot^{2}{\left(5 x \right)} + 5 \cot^{2}{\left(3 x \right)}}{\left(3 \cot^{2}{\left(3 x \right)} + 3\right) \cot^{2}{\left(5 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{3 \cot^{2}{\left(3 x \right)} + 3}}{\frac{d}{d x} \frac{\cot^{2}{\left(5 x \right)}}{5 \cot^{2}{\left(3 x \right)} \cot^{2}{\left(5 x \right)} + 5 \cot^{2}{\left(3 x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{3 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)}}{\left(\frac{\left(- 10 \cot^{2}{\left(5 x \right)} - 10\right) \cot{\left(5 x \right)}}{5 \cot^{2}{\left(3 x \right)} \cot^{2}{\left(5 x \right)} + 5 \cot^{2}{\left(3 x \right)}} + \frac{\left(- 5 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)} \cot^{2}{\left(5 x \right)} - 5 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)} - 5 \left(- 10 \cot^{2}{\left(5 x \right)} - 10\right) \cot^{2}{\left(3 x \right)} \cot{\left(5 x \right)}\right) \cot^{2}{\left(5 x \right)}}{\left(5 \cot^{2}{\left(3 x \right)} \cot^{2}{\left(5 x \right)} + 5 \cot^{2}{\left(3 x \right)}\right)^{2}}\right) \left(3 \cot^{2}{\left(3 x \right)} + 3\right)^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{3 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)}}{\left(\frac{\left(- 10 \cot^{2}{\left(5 x \right)} - 10\right) \cot{\left(5 x \right)}}{5 \cot^{2}{\left(3 x \right)} \cot^{2}{\left(5 x \right)} + 5 \cot^{2}{\left(3 x \right)}} + \frac{\left(- 5 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)} \cot^{2}{\left(5 x \right)} - 5 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)} - 5 \left(- 10 \cot^{2}{\left(5 x \right)} - 10\right) \cot^{2}{\left(3 x \right)} \cot{\left(5 x \right)}\right) \cot^{2}{\left(5 x \right)}}{\left(5 \cot^{2}{\left(3 x \right)} \cot^{2}{\left(5 x \right)} + 5 \cot^{2}{\left(3 x \right)}\right)^{2}}\right) \left(3 \cot^{2}{\left(3 x \right)} + 3\right)^{2}}\right)$$
=
$$\frac{5}{3}$$
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
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(5 x \right)}} = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{\cot{\left(3 x \right)}}{\cot{\left(5 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot{\left(5 x \right)}}}{\frac{d}{d x} \frac{1}{\cot{\left(3 x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5 \cot^{2}{\left(3 x \right)} \cot^{2}{\left(5 x \right)} + 5 \cot^{2}{\left(3 x \right)}}{\left(3 \cot^{2}{\left(3 x \right)} + 3\right) \cot^{2}{\left(5 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{3 \cot^{2}{\left(3 x \right)} + 3}}{\frac{d}{d x} \frac{\cot^{2}{\left(5 x \right)}}{5 \cot^{2}{\left(3 x \right)} \cot^{2}{\left(5 x \right)} + 5 \cot^{2}{\left(3 x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{3 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)}}{\left(\frac{\left(- 10 \cot^{2}{\left(5 x \right)} - 10\right) \cot{\left(5 x \right)}}{5 \cot^{2}{\left(3 x \right)} \cot^{2}{\left(5 x \right)} + 5 \cot^{2}{\left(3 x \right)}} + \frac{\left(- 5 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)} \cot^{2}{\left(5 x \right)} - 5 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)} - 5 \left(- 10 \cot^{2}{\left(5 x \right)} - 10\right) \cot^{2}{\left(3 x \right)} \cot{\left(5 x \right)}\right) \cot^{2}{\left(5 x \right)}}{\left(5 \cot^{2}{\left(3 x \right)} \cot^{2}{\left(5 x \right)} + 5 \cot^{2}{\left(3 x \right)}\right)^{2}}\right) \left(3 \cot^{2}{\left(3 x \right)} + 3\right)^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{3 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)}}{\left(\frac{\left(- 10 \cot^{2}{\left(5 x \right)} - 10\right) \cot{\left(5 x \right)}}{5 \cot^{2}{\left(3 x \right)} \cot^{2}{\left(5 x \right)} + 5 \cot^{2}{\left(3 x \right)}} + \frac{\left(- 5 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)} \cot^{2}{\left(5 x \right)} - 5 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)} - 5 \left(- 10 \cot^{2}{\left(5 x \right)} - 10\right) \cot^{2}{\left(3 x \right)} \cot{\left(5 x \right)}\right) \cot^{2}{\left(5 x \right)}}{\left(5 \cot^{2}{\left(3 x \right)} \cot^{2}{\left(5 x \right)} + 5 \cot^{2}{\left(3 x \right)}\right)^{2}}\right) \left(3 \cot^{2}{\left(3 x \right)} + 3\right)^{2}}\right)$$
=
$$\frac{5}{3}$$
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]
/cot(3*x)\ lim |--------| x->0+\cot(5*x)/
$$\lim_{x \to 0^+}\left(\frac{\cot{\left(3 x \right)}}{\cot{\left(5 x \right)}}\right)$$
5/3
$$\frac{5}{3}$$
= 1.66666666666667
/cot(3*x)\ lim |--------| x->0-\cot(5*x)/
$$\lim_{x \to 0^-}\left(\frac{\cot{\left(3 x \right)}}{\cot{\left(5 x \right)}}\right)$$
5/3
$$\frac{5}{3}$$
= 1.66666666666667
= 1.66666666666667
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\cot{\left(3 x \right)}}{\cot{\left(5 x \right)}}\right) = \frac{5}{3}$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\cot{\left(3 x \right)}}{\cot{\left(5 x \right)}}\right) = \frac{5}{3}$$
$$\lim_{x \to \infty}\left(\frac{\cot{\left(3 x \right)}}{\cot{\left(5 x \right)}}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\cot{\left(3 x \right)}}{\cot{\left(5 x \right)}}\right) = \frac{\tan{\left(5 \right)}}{\tan{\left(3 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\cot{\left(3 x \right)}}{\cot{\left(5 x \right)}}\right) = \frac{\tan{\left(5 \right)}}{\tan{\left(3 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\cot{\left(3 x \right)}}{\cot{\left(5 x \right)}}\right)$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\cot{\left(3 x \right)}}{\cot{\left(5 x \right)}}\right) = \frac{5}{3}$$
$$\lim_{x \to \infty}\left(\frac{\cot{\left(3 x \right)}}{\cot{\left(5 x \right)}}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\cot{\left(3 x \right)}}{\cot{\left(5 x \right)}}\right) = \frac{\tan{\left(5 \right)}}{\tan{\left(3 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\cot{\left(3 x \right)}}{\cot{\left(5 x \right)}}\right) = \frac{\tan{\left(5 \right)}}{\tan{\left(3 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\cot{\left(3 x \right)}}{\cot{\left(5 x \right)}}\right)$$
Más detalles con x→-oo
Gráfico