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- Límite de la función:
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Límite de (-1+sqrt(1+x))/(-1+(1+x)^(1/3))
-
Límite de (-cos(x)+cos(3*x))/(-1+cos(x))
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Límite de -1/2+9*x
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Límite de (4+5*x+6*x^2)/(-2+3*x^2+7*x)
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Expresiones idénticas
- tan(cinco *x)/(dos +tan(x))
- tangente de (5 multiplicar por x) dividir por (2 más tangente de (x))
- tangente de (cinco multiplicar por x) dividir por (dos más tangente de (x))
- tan(5x)/(2+tan(x))
- tan5x/2+tanx
- tan(5*x) dividir por (2+tan(x))
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Expresiones semejantes
- tan(5*x)/(2-tan(x))
-
Expresiones con funciones
- Tangente tan
- tan(9*x)/3
- tan(5*sqrt(x))^2/sin(25*x)
- tan(2*x)/(-1+cos(3*x))
- tan(x)^3*log(-1+exp(5*x))/(1-cos(x^2))
- tan(5^(-1-x))/((2+4*x)*tan(5^(-x)))
- Tangente tan
- tan(9*x)/3
- tan(5*sqrt(x))^2/sin(25*x)
- tan(2*x)/(-1+cos(3*x))
- tan(x)^3*log(-1+exp(5*x))/(1-cos(x^2))
- tan(5^(-1-x))/((2+4*x)*tan(5^(-x)))
Límite de la función tan(5*x)/(2+tan(x))
Solución
Ha introducido
[src]
/ tan(5*x) \ lim |----------| pi \2 + tan(x)/ x->--+ 2
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\tan{\left(5 x \right)}}{\tan{\left(x \right)} + 2}\right)$$
Limit(tan(5*x)/(2 + tan(x)), x, pi/2)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to \frac{\pi}{2}^+} \frac{1}{\tan{\left(x \right)} + 2} = 0$$
y el límite para el denominador es
$$\lim_{x \to \frac{\pi}{2}^+} \frac{1}{\tan{\left(5 x \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\tan{\left(5 x \right)}}{\tan{\left(x \right)} + 2}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{1}{\tan{\left(x \right)} + 2}}{\frac{d}{d x} \frac{1}{\tan{\left(5 x \right)}}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{1}{\left(- 5 \tan^{2}{\left(5 x \right)} - 5\right) \left(\frac{\tan^{2}{\left(x \right)}}{- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}} + \frac{4 \tan{\left(x \right)}}{- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}} + \frac{4}{- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}}\right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{1}{- 5 \tan^{2}{\left(5 x \right)} - 5}}{\frac{d}{d x} \left(\frac{\tan^{2}{\left(x \right)}}{- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}} + \frac{4 \tan{\left(x \right)}}{- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}} + \frac{4}{- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}}\right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{5 \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan{\left(5 x \right)}}{\left(- 5 \tan^{2}{\left(5 x \right)} - 5\right)^{2} \left(\frac{4 \left(\tan^{2}{\left(x \right)} + 1\right)}{- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}} + \frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}}{- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}} + \frac{\left(\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} \tan^{2}{\left(5 x \right)} + \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan^{2}{\left(x \right)} \tan{\left(5 x \right)} + \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan{\left(5 x \right)}\right) \tan^{2}{\left(x \right)}}{\left(- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}\right)^{2}} + \frac{4 \left(\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} \tan^{2}{\left(5 x \right)} + \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan^{2}{\left(x \right)} \tan{\left(5 x \right)} + \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan{\left(5 x \right)}\right) \tan{\left(x \right)}}{\left(- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}\right)^{2}} + \frac{4 \left(\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} \tan^{2}{\left(5 x \right)} + \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan^{2}{\left(x \right)} \tan{\left(5 x \right)} + \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan{\left(5 x \right)}\right)}{\left(- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}\right)^{2}}\right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{5 \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan{\left(5 x \right)}}{\left(- 5 \tan^{2}{\left(5 x \right)} - 5\right)^{2} \left(\frac{4 \left(\tan^{2}{\left(x \right)} + 1\right)}{- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}} + \frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}}{- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}} + \frac{\left(\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} \tan^{2}{\left(5 x \right)} + \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan^{2}{\left(x \right)} \tan{\left(5 x \right)} + \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan{\left(5 x \right)}\right) \tan^{2}{\left(x \right)}}{\left(- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}\right)^{2}} + \frac{4 \left(\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} \tan^{2}{\left(5 x \right)} + \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan^{2}{\left(x \right)} \tan{\left(5 x \right)} + \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan{\left(5 x \right)}\right) \tan{\left(x \right)}}{\left(- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}\right)^{2}} + \frac{4 \left(\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} \tan^{2}{\left(5 x \right)} + \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan^{2}{\left(x \right)} \tan{\left(5 x \right)} + \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan{\left(5 x \right)}\right)}{\left(- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}\right)^{2}}\right)}\right)$$
=
$$\frac{1}{5}$$
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
$$\lim_{x \to \frac{\pi}{2}^+} \frac{1}{\tan{\left(x \right)} + 2} = 0$$
y el límite para el denominador es
$$\lim_{x \to \frac{\pi}{2}^+} \frac{1}{\tan{\left(5 x \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\tan{\left(5 x \right)}}{\tan{\left(x \right)} + 2}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{1}{\tan{\left(x \right)} + 2}}{\frac{d}{d x} \frac{1}{\tan{\left(5 x \right)}}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{1}{\left(- 5 \tan^{2}{\left(5 x \right)} - 5\right) \left(\frac{\tan^{2}{\left(x \right)}}{- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}} + \frac{4 \tan{\left(x \right)}}{- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}} + \frac{4}{- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}}\right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{1}{- 5 \tan^{2}{\left(5 x \right)} - 5}}{\frac{d}{d x} \left(\frac{\tan^{2}{\left(x \right)}}{- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}} + \frac{4 \tan{\left(x \right)}}{- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}} + \frac{4}{- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}}\right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{5 \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan{\left(5 x \right)}}{\left(- 5 \tan^{2}{\left(5 x \right)} - 5\right)^{2} \left(\frac{4 \left(\tan^{2}{\left(x \right)} + 1\right)}{- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}} + \frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}}{- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}} + \frac{\left(\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} \tan^{2}{\left(5 x \right)} + \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan^{2}{\left(x \right)} \tan{\left(5 x \right)} + \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan{\left(5 x \right)}\right) \tan^{2}{\left(x \right)}}{\left(- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}\right)^{2}} + \frac{4 \left(\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} \tan^{2}{\left(5 x \right)} + \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan^{2}{\left(x \right)} \tan{\left(5 x \right)} + \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan{\left(5 x \right)}\right) \tan{\left(x \right)}}{\left(- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}\right)^{2}} + \frac{4 \left(\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} \tan^{2}{\left(5 x \right)} + \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan^{2}{\left(x \right)} \tan{\left(5 x \right)} + \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan{\left(5 x \right)}\right)}{\left(- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}\right)^{2}}\right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{5 \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan{\left(5 x \right)}}{\left(- 5 \tan^{2}{\left(5 x \right)} - 5\right)^{2} \left(\frac{4 \left(\tan^{2}{\left(x \right)} + 1\right)}{- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}} + \frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}}{- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}} + \frac{\left(\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} \tan^{2}{\left(5 x \right)} + \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan^{2}{\left(x \right)} \tan{\left(5 x \right)} + \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan{\left(5 x \right)}\right) \tan^{2}{\left(x \right)}}{\left(- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}\right)^{2}} + \frac{4 \left(\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} \tan^{2}{\left(5 x \right)} + \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan^{2}{\left(x \right)} \tan{\left(5 x \right)} + \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan{\left(5 x \right)}\right) \tan{\left(x \right)}}{\left(- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}\right)^{2}} + \frac{4 \left(\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} \tan^{2}{\left(5 x \right)} + \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan^{2}{\left(x \right)} \tan{\left(5 x \right)} + \left(10 \tan^{2}{\left(5 x \right)} + 10\right) \tan{\left(5 x \right)}\right)}{\left(- \tan^{2}{\left(x \right)} \tan^{2}{\left(5 x \right)} - \tan^{2}{\left(5 x \right)}\right)^{2}}\right)}\right)$$
=
$$\frac{1}{5}$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 2 vez (veces)
Gráfica
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to \frac{\pi}{2}^-}\left(\frac{\tan{\left(5 x \right)}}{\tan{\left(x \right)} + 2}\right) = \frac{1}{5}$$
Más detalles con x→pi/2 a la izquierda
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\tan{\left(5 x \right)}}{\tan{\left(x \right)} + 2}\right) = \frac{1}{5}$$
$$\lim_{x \to \infty}\left(\frac{\tan{\left(5 x \right)}}{\tan{\left(x \right)} + 2}\right)$$
Más detalles con x→oo
$$\lim_{x \to 0^-}\left(\frac{\tan{\left(5 x \right)}}{\tan{\left(x \right)} + 2}\right) = 0$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(5 x \right)}}{\tan{\left(x \right)} + 2}\right) = 0$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{\tan{\left(5 x \right)}}{\tan{\left(x \right)} + 2}\right) = \frac{\tan{\left(5 \right)}}{\tan{\left(1 \right)} + 2}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\tan{\left(5 x \right)}}{\tan{\left(x \right)} + 2}\right) = \frac{\tan{\left(5 \right)}}{\tan{\left(1 \right)} + 2}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(5 x \right)}}{\tan{\left(x \right)} + 2}\right)$$
Más detalles con x→-oo
Más detalles con x→pi/2 a la izquierda
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\tan{\left(5 x \right)}}{\tan{\left(x \right)} + 2}\right) = \frac{1}{5}$$
$$\lim_{x \to \infty}\left(\frac{\tan{\left(5 x \right)}}{\tan{\left(x \right)} + 2}\right)$$
Más detalles con x→oo
$$\lim_{x \to 0^-}\left(\frac{\tan{\left(5 x \right)}}{\tan{\left(x \right)} + 2}\right) = 0$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(5 x \right)}}{\tan{\left(x \right)} + 2}\right) = 0$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{\tan{\left(5 x \right)}}{\tan{\left(x \right)} + 2}\right) = \frac{\tan{\left(5 \right)}}{\tan{\left(1 \right)} + 2}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\tan{\left(5 x \right)}}{\tan{\left(x \right)} + 2}\right) = \frac{\tan{\left(5 \right)}}{\tan{\left(1 \right)} + 2}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(5 x \right)}}{\tan{\left(x \right)} + 2}\right)$$
Más detalles con x→-oo
A la izquierda y a la derecha
[src]
/ tan(5*x) \ lim |----------| pi \2 + tan(x)/ x->--+ 2
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\tan{\left(5 x \right)}}{\tan{\left(x \right)} + 2}\right)$$
1/5
$$\frac{1}{5}$$
= 0.2
/ tan(5*x) \ lim |----------| pi \2 + tan(x)/ x->--- 2
$$\lim_{x \to \frac{\pi}{2}^-}\left(\frac{\tan{\left(5 x \right)}}{\tan{\left(x \right)} + 2}\right)$$
1/5
$$\frac{1}{5}$$
= 0.2
= 0.2