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- Límite de la función:
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Límite de (-5+x)/(-25+x^2)
-
Límite de (-1+e^x)/x
-
Límite de (x^2-2*x)/(4+x^2-4*x)
-
Límite de (1+x)^(1/x)
-
Expresiones idénticas
- -atan(tres *x)*cot(x)
- menos arco tangente de gente de (3 multiplicar por x) multiplicar por cotangente de (x)
- menos arco tangente de gente de (tres multiplicar por x) multiplicar por cotangente de (x)
- -atan(3x)cot(x)
- -atan3xcotx
-
Expresiones semejantes
- atan(3*x)*cot(x)
- -arctan(3*x)*cot(x)
-
Expresiones con funciones
- Arcotangente arctan
- atan(x)/x
- atan(log(x))
- atan(3*x)/(5*x)
- atan(3*x)/asin(2*x)
- atan(5*x)/(3*x)
- Cotangente cot
- cot(x/2)^(1/cos(x))
- cot(2*x)*sin(6*x)
- cot(5*pi*x)*log(x)
- cot(x/4)^(1/cos(x/2))
- cot(2*x)/(5*x)
Límite de la función -atan(3*x)*cot(x)
Solución
Ha introducido
[src]
lim (-atan(3*x)*cot(x)) x->0+
$$\lim_{x \to 0^+}\left(\cot{\left(x \right)} \left(- \operatorname{atan}{\left(3 x \right)}\right)\right)$$
Limit((-atan(3*x))*cot(x), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+} \operatorname{atan}{\left(3 x \right)} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+}\left(- \frac{1}{\cot{\left(x \right)}}\right) = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\cot{\left(x \right)} \left(- \operatorname{atan}{\left(3 x \right)}\right)\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(- \cot{\left(x \right)} \operatorname{atan}{\left(3 x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \operatorname{atan}{\left(3 x \right)}}{\frac{d}{d x} \left(- \frac{1}{\cot{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{3 \cot^{2}{\left(x \right)}}{\left(9 x^{2} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{3 \cot^{2}{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot^{2}{\left(x \right)} + 1}}{\frac{d}{d x} \left(- \frac{1}{3 \cot^{2}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\left(2 \cot^{2}{\left(x \right)} + 2\right) \left(\frac{\cot^{4}{\left(x \right)}}{- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}} + \frac{2 \cot^{2}{\left(x \right)}}{- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}} + \frac{1}{- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(x \right)} + 2}}{\frac{d}{d x} \left(\frac{\cot^{4}{\left(x \right)}}{- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}} + \frac{2 \cot^{2}{\left(x \right)}}{- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}} + \frac{1}{- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(\frac{\left(6 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 6 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}\right)^{2}} + \frac{2 \left(6 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 6 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}\right)^{2}} + \frac{6 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 6 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{\left(- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}\right)^{2}} + \frac{\left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}} + \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(\frac{\left(6 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 6 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}\right)^{2}} + \frac{2 \left(6 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 6 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}\right)^{2}} + \frac{6 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 6 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{\left(- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}\right)^{2}} + \frac{\left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}} + \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$-3$$
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
$$\lim_{x \to 0^+} \operatorname{atan}{\left(3 x \right)} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+}\left(- \frac{1}{\cot{\left(x \right)}}\right) = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\cot{\left(x \right)} \left(- \operatorname{atan}{\left(3 x \right)}\right)\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(- \cot{\left(x \right)} \operatorname{atan}{\left(3 x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \operatorname{atan}{\left(3 x \right)}}{\frac{d}{d x} \left(- \frac{1}{\cot{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{3 \cot^{2}{\left(x \right)}}{\left(9 x^{2} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{3 \cot^{2}{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot^{2}{\left(x \right)} + 1}}{\frac{d}{d x} \left(- \frac{1}{3 \cot^{2}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\left(2 \cot^{2}{\left(x \right)} + 2\right) \left(\frac{\cot^{4}{\left(x \right)}}{- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}} + \frac{2 \cot^{2}{\left(x \right)}}{- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}} + \frac{1}{- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(x \right)} + 2}}{\frac{d}{d x} \left(\frac{\cot^{4}{\left(x \right)}}{- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}} + \frac{2 \cot^{2}{\left(x \right)}}{- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}} + \frac{1}{- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(\frac{\left(6 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 6 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}\right)^{2}} + \frac{2 \left(6 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 6 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}\right)^{2}} + \frac{6 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 6 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{\left(- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}\right)^{2}} + \frac{\left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}} + \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(\frac{\left(6 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 6 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}\right)^{2}} + \frac{2 \left(6 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 6 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}\right)^{2}} + \frac{6 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 6 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{\left(- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}\right)^{2}} + \frac{\left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}} + \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 6 \cot^{6}{\left(x \right)} - 6 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$-3$$
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
$$\lim_{x \to 0^-}\left(\cot{\left(x \right)} \left(- \operatorname{atan}{\left(3 x \right)}\right)\right) = -3$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\cot{\left(x \right)} \left(- \operatorname{atan}{\left(3 x \right)}\right)\right) = -3$$
$$\lim_{x \to \infty}\left(\cot{\left(x \right)} \left(- \operatorname{atan}{\left(3 x \right)}\right)\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\cot{\left(x \right)} \left(- \operatorname{atan}{\left(3 x \right)}\right)\right) = - \frac{\operatorname{atan}{\left(3 \right)}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\cot{\left(x \right)} \left(- \operatorname{atan}{\left(3 x \right)}\right)\right) = - \frac{\operatorname{atan}{\left(3 \right)}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\cot{\left(x \right)} \left(- \operatorname{atan}{\left(3 x \right)}\right)\right)$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\cot{\left(x \right)} \left(- \operatorname{atan}{\left(3 x \right)}\right)\right) = -3$$
$$\lim_{x \to \infty}\left(\cot{\left(x \right)} \left(- \operatorname{atan}{\left(3 x \right)}\right)\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\cot{\left(x \right)} \left(- \operatorname{atan}{\left(3 x \right)}\right)\right) = - \frac{\operatorname{atan}{\left(3 \right)}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\cot{\left(x \right)} \left(- \operatorname{atan}{\left(3 x \right)}\right)\right) = - \frac{\operatorname{atan}{\left(3 \right)}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\cot{\left(x \right)} \left(- \operatorname{atan}{\left(3 x \right)}\right)\right)$$
Más detalles con x→-oo
A la izquierda y a la derecha
[src]
lim (-atan(3*x)*cot(x)) x->0+
$$\lim_{x \to 0^+}\left(\cot{\left(x \right)} \left(- \operatorname{atan}{\left(3 x \right)}\right)\right)$$
-3
$$-3$$
= -3
lim (-atan(3*x)*cot(x)) x->0-
$$\lim_{x \to 0^-}\left(\cot{\left(x \right)} \left(- \operatorname{atan}{\left(3 x \right)}\right)\right)$$
-3
$$-3$$
= -3
= -3