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(1-e^(2*x))*cot(x)

Límite de la función (1-e^(2*x))*cot(x)

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Solución

Ha introducido [src]
     //     2*x\       \
 lim \\1 - E   /*cot(x)/
x->0+                   
$$\lim_{x \to 0^+}\left(\left(1 - e^{2 x}\right) \cot{\left(x \right)}\right)$$
Limit((1 - E^(2*x))*cot(x), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
0/0,

tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(1 - e^{2 x}\right) = 0$$
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.
$$\lim_{x \to 0^+}\left(\left(1 - e^{2 x}\right) \cot{\left(x \right)}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\left(1 - e^{2 x}\right) \cot{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(1 - e^{2 x}\right)}{\frac{d}{d x} \frac{1}{\cot{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 e^{2 x} \cot^{2}{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \cot^{2}{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot^{2}{\left(x \right)} + 1}}{\frac{d}{d x} \left(- \frac{1}{2 \cot^{2}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\left(2 \cot^{2}{\left(x \right)} + 2\right) \left(\frac{\cot^{4}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} + \frac{2 \cot^{2}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} + \frac{1}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(x \right)} + 2}}{\frac{d}{d x} \left(\frac{\cot^{4}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} + \frac{2 \cot^{2}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} + \frac{1}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(\frac{\left(4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} + \frac{2 \left(4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} + \frac{4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} + \frac{\left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} + \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(\frac{\left(4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} + \frac{2 \left(4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} + \frac{4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} + \frac{\left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} + \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$-2$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)
Gráfica
Respuesta rápida [src]
-2
$$-2$$
A la izquierda y a la derecha [src]
     //     2*x\       \
 lim \\1 - E   /*cot(x)/
x->0+                   
$$\lim_{x \to 0^+}\left(\left(1 - e^{2 x}\right) \cot{\left(x \right)}\right)$$
-2
$$-2$$
= -2.0
     //     2*x\       \
 lim \\1 - E   /*cot(x)/
x->0-                   
$$\lim_{x \to 0^-}\left(\left(1 - e^{2 x}\right) \cot{\left(x \right)}\right)$$
-2
$$-2$$
= -2.0
= -2.0
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\left(1 - e^{2 x}\right) \cot{\left(x \right)}\right) = -2$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\left(1 - e^{2 x}\right) \cot{\left(x \right)}\right) = -2$$
$$\lim_{x \to \infty}\left(\left(1 - e^{2 x}\right) \cot{\left(x \right)}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\left(1 - e^{2 x}\right) \cot{\left(x \right)}\right) = - \frac{-1 + e^{2}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\left(1 - e^{2 x}\right) \cot{\left(x \right)}\right) = - \frac{-1 + e^{2}}{\tan{\left(1 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\left(1 - e^{2 x}\right) \cot{\left(x \right)}\right)$$
Más detalles con x→-oo
Respuesta numérica [src]
-2.0
-2.0
Gráfico
Límite de la función (1-e^(2*x))*cot(x)