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Límite de 1/(1-x)-3/(1-x^3)
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Límite de sin(3*x)/(2*x)
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Límite de (1-2*x)^(1/x)
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Límite de (6+x^2-5*x)/(-9+x^2)
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Expresiones idénticas
- tan(x)/log(pi/ dos -x)
- tangente de (x) dividir por logaritmo de ( número pi dividir por 2 menos x)
- tangente de (x) dividir por logaritmo de ( número pi dividir por dos menos x)
- tanx/logpi/2-x
- tan(x) dividir por log(pi dividir por 2-x)
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Expresiones semejantes
- tan(x)/log(pi/2+x)
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Expresiones con funciones
- Tangente tan
- tan(5*x)/sin(3*x)
- tan(4*x)/x
- tan(x/2)^2/x^2
- tan(x)^2
- tan(5*x)/tan(3*x)
- Logaritmo log
- log(x)/log(1+x)
- log(log(x))
- log(sin(m*x))/log(sin(x))
- log(1+3*x)/x
- log(x)/(-1+x^2)
- Número Pi pi
- Piecewise((x,x<0),(-x,x<2),(0,True))
- pi*x*sin(pi/x)^2
- Piecewise((3+x^2+5*x,x<=-2),(-2+x^2,True))
- pi^4*sin(x)/((pi+x)*(pi-x))
- pi/4-x*atan(x/(1+x))
Límite de la función tan(x)/log(pi/2-x)
Solución
Ha introducido
[src]
/ tan(x) \ lim |-----------| pi | /pi \| x->--+|log|-- - x|| 2 \ \2 //
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\tan{\left(x \right)}}{\log{\left(- x + \frac{\pi}{2} \right)}}\right)$$
Limit(tan(x)/log(pi/2 - x), x, pi/2)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to \frac{\pi}{2}^+} \frac{1}{\log{\left(\pi - 2 x \right)} - \log{\left(2 \right)}} = 0$$
y el límite para el denominador es
$$\lim_{x \to \frac{\pi}{2}^+} \frac{1}{\tan{\left(x \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\tan{\left(x \right)}}{\log{\left(- x + \frac{\pi}{2} \right)}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\tan{\left(x \right)}}{\log{\left(\frac{\pi - 2 x}{2} \right)}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(\pi - 2 x \right)} - \log{\left(2 \right)}}}{\frac{d}{d x} \frac{1}{\tan{\left(x \right)}}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{1}{\left(\log{\left(\pi - 2 x \right)}^{2} - 2 \log{\left(2 \right)} \log{\left(\pi - 2 x \right)} + \log{\left(2 \right)}^{2}\right) \left(x + \frac{x}{\tan^{2}{\left(x \right)}} - \frac{\pi}{2} - \frac{\pi}{2 \tan^{2}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(\pi - 2 x \right)}^{2} - 2 \log{\left(2 \right)} \log{\left(\pi - 2 x \right)} + \log{\left(2 \right)}^{2}}}{\frac{d}{d x} \left(x + \frac{x}{\tan^{2}{\left(x \right)}} - \frac{\pi}{2} - \frac{\pi}{2 \tan^{2}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{1}{\left(\frac{\log{\left(\pi - 2 x \right)}^{4}}{\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}} - \frac{4 \log{\left(2 \right)} \log{\left(\pi - 2 x \right)}^{3}}{\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}} + \frac{6 \log{\left(2 \right)}^{2} \log{\left(\pi - 2 x \right)}^{2}}{\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}} - \frac{4 \log{\left(2 \right)}^{3} \log{\left(\pi - 2 x \right)}}{\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}} + \frac{\log{\left(2 \right)}^{4}}{\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}}\right) \left(- \frac{2 x}{\tan{\left(x \right)}} - \frac{2 x}{\tan^{3}{\left(x \right)}} + 1 + \frac{\pi}{\tan{\left(x \right)}} + \frac{1}{\tan^{2}{\left(x \right)}} + \frac{\pi}{\tan^{3}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{1}{- \frac{2 x}{\tan{\left(x \right)}} - \frac{2 x}{\tan^{3}{\left(x \right)}} + 1 + \frac{\pi}{\tan{\left(x \right)}} + \frac{1}{\tan^{2}{\left(x \right)}} + \frac{\pi}{\tan^{3}{\left(x \right)}}}}{\frac{d}{d x} \left(\frac{\log{\left(\pi - 2 x \right)}^{4}}{\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}} - \frac{4 \log{\left(2 \right)} \log{\left(\pi - 2 x \right)}^{3}}{\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}} + \frac{6 \log{\left(2 \right)}^{2} \log{\left(\pi - 2 x \right)}^{2}}{\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}} - \frac{4 \log{\left(2 \right)}^{3} \log{\left(\pi - 2 x \right)}}{\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}} + \frac{\log{\left(2 \right)}^{4}}{\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}}\right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{2 x \left(- 3 \tan^{2}{\left(x \right)} - 3\right)}{\tan^{4}{\left(x \right)}} + \frac{2 x \left(- \tan^{2}{\left(x \right)} - 1\right)}{\tan^{2}{\left(x \right)}} - \frac{\pi \left(- 3 \tan^{2}{\left(x \right)} - 3\right)}{\tan^{4}{\left(x \right)}} - \frac{- 2 \tan^{2}{\left(x \right)} - 2}{\tan^{3}{\left(x \right)}} - \frac{\pi \left(- \tan^{2}{\left(x \right)} - 1\right)}{\tan^{2}{\left(x \right)}} + \frac{2}{\tan{\left(x \right)}} + \frac{2}{\tan^{3}{\left(x \right)}}}{\left(- \frac{2 x}{\tan{\left(x \right)}} - \frac{2 x}{\tan^{3}{\left(x \right)}} + 1 + \frac{\pi}{\tan{\left(x \right)}} + \frac{1}{\tan^{2}{\left(x \right)}} + \frac{\pi}{\tan^{3}{\left(x \right)}}\right)^{2} \left(\frac{\left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(\pi - 2 x \right)}^{4}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} - \frac{4 \left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(2 \right)} \log{\left(\pi - 2 x \right)}^{3}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} + \frac{6 \left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(2 \right)}^{2} \log{\left(\pi - 2 x \right)}^{2}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} - \frac{4 \left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(2 \right)}^{3} \log{\left(\pi - 2 x \right)}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} + \frac{\left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(2 \right)}^{4}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} - \frac{8 \log{\left(\pi - 2 x \right)}^{3}}{\left(\pi - 2 x\right) \left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)} + \frac{24 \log{\left(2 \right)} \log{\left(\pi - 2 x \right)}^{2}}{\left(\pi - 2 x\right) \left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)} - \frac{24 \log{\left(2 \right)}^{2} \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right) \left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)} + \frac{8 \log{\left(2 \right)}^{3}}{\left(\pi - 2 x\right) \left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)}\right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{2 x \left(- 3 \tan^{2}{\left(x \right)} - 3\right)}{\tan^{4}{\left(x \right)}} + \frac{2 x \left(- \tan^{2}{\left(x \right)} - 1\right)}{\tan^{2}{\left(x \right)}} - \frac{\pi \left(- 3 \tan^{2}{\left(x \right)} - 3\right)}{\tan^{4}{\left(x \right)}} - \frac{- 2 \tan^{2}{\left(x \right)} - 2}{\tan^{3}{\left(x \right)}} - \frac{\pi \left(- \tan^{2}{\left(x \right)} - 1\right)}{\tan^{2}{\left(x \right)}} + \frac{2}{\tan{\left(x \right)}} + \frac{2}{\tan^{3}{\left(x \right)}}}{\frac{\left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(\pi - 2 x \right)}^{4}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} - \frac{4 \left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(2 \right)} \log{\left(\pi - 2 x \right)}^{3}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} + \frac{6 \left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(2 \right)}^{2} \log{\left(\pi - 2 x \right)}^{2}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} - \frac{4 \left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(2 \right)}^{3} \log{\left(\pi - 2 x \right)}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} + \frac{\left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(2 \right)}^{4}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} - \frac{8 \log{\left(\pi - 2 x \right)}^{3}}{\left(\pi - 2 x\right) \left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)} + \frac{24 \log{\left(2 \right)} \log{\left(\pi - 2 x \right)}^{2}}{\left(\pi - 2 x\right) \left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)} - \frac{24 \log{\left(2 \right)}^{2} \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right) \left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)} + \frac{8 \log{\left(2 \right)}^{3}}{\left(\pi - 2 x\right) \left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{2 x \left(- 3 \tan^{2}{\left(x \right)} - 3\right)}{\tan^{4}{\left(x \right)}} + \frac{2 x \left(- \tan^{2}{\left(x \right)} - 1\right)}{\tan^{2}{\left(x \right)}} - \frac{\pi \left(- 3 \tan^{2}{\left(x \right)} - 3\right)}{\tan^{4}{\left(x \right)}} - \frac{- 2 \tan^{2}{\left(x \right)} - 2}{\tan^{3}{\left(x \right)}} - \frac{\pi \left(- \tan^{2}{\left(x \right)} - 1\right)}{\tan^{2}{\left(x \right)}} + \frac{2}{\tan{\left(x \right)}} + \frac{2}{\tan^{3}{\left(x \right)}}}{\frac{\left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(\pi - 2 x \right)}^{4}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} - \frac{4 \left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(2 \right)} \log{\left(\pi - 2 x \right)}^{3}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} + \frac{6 \left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(2 \right)}^{2} \log{\left(\pi - 2 x \right)}^{2}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} - \frac{4 \left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(2 \right)}^{3} \log{\left(\pi - 2 x \right)}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} + \frac{\left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(2 \right)}^{4}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} - \frac{8 \log{\left(\pi - 2 x \right)}^{3}}{\left(\pi - 2 x\right) \left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)} + \frac{24 \log{\left(2 \right)} \log{\left(\pi - 2 x \right)}^{2}}{\left(\pi - 2 x\right) \left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)} - \frac{24 \log{\left(2 \right)}^{2} \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right) \left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)} + \frac{8 \log{\left(2 \right)}^{3}}{\left(\pi - 2 x\right) \left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)}}\right)$$
=
$$\infty$$
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
$$\lim_{x \to \frac{\pi}{2}^+} \frac{1}{\log{\left(\pi - 2 x \right)} - \log{\left(2 \right)}} = 0$$
y el límite para el denominador es
$$\lim_{x \to \frac{\pi}{2}^+} \frac{1}{\tan{\left(x \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\tan{\left(x \right)}}{\log{\left(- x + \frac{\pi}{2} \right)}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\tan{\left(x \right)}}{\log{\left(\frac{\pi - 2 x}{2} \right)}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(\pi - 2 x \right)} - \log{\left(2 \right)}}}{\frac{d}{d x} \frac{1}{\tan{\left(x \right)}}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{1}{\left(\log{\left(\pi - 2 x \right)}^{2} - 2 \log{\left(2 \right)} \log{\left(\pi - 2 x \right)} + \log{\left(2 \right)}^{2}\right) \left(x + \frac{x}{\tan^{2}{\left(x \right)}} - \frac{\pi}{2} - \frac{\pi}{2 \tan^{2}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(\pi - 2 x \right)}^{2} - 2 \log{\left(2 \right)} \log{\left(\pi - 2 x \right)} + \log{\left(2 \right)}^{2}}}{\frac{d}{d x} \left(x + \frac{x}{\tan^{2}{\left(x \right)}} - \frac{\pi}{2} - \frac{\pi}{2 \tan^{2}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{1}{\left(\frac{\log{\left(\pi - 2 x \right)}^{4}}{\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}} - \frac{4 \log{\left(2 \right)} \log{\left(\pi - 2 x \right)}^{3}}{\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}} + \frac{6 \log{\left(2 \right)}^{2} \log{\left(\pi - 2 x \right)}^{2}}{\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}} - \frac{4 \log{\left(2 \right)}^{3} \log{\left(\pi - 2 x \right)}}{\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}} + \frac{\log{\left(2 \right)}^{4}}{\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}}\right) \left(- \frac{2 x}{\tan{\left(x \right)}} - \frac{2 x}{\tan^{3}{\left(x \right)}} + 1 + \frac{\pi}{\tan{\left(x \right)}} + \frac{1}{\tan^{2}{\left(x \right)}} + \frac{\pi}{\tan^{3}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{1}{- \frac{2 x}{\tan{\left(x \right)}} - \frac{2 x}{\tan^{3}{\left(x \right)}} + 1 + \frac{\pi}{\tan{\left(x \right)}} + \frac{1}{\tan^{2}{\left(x \right)}} + \frac{\pi}{\tan^{3}{\left(x \right)}}}}{\frac{d}{d x} \left(\frac{\log{\left(\pi - 2 x \right)}^{4}}{\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}} - \frac{4 \log{\left(2 \right)} \log{\left(\pi - 2 x \right)}^{3}}{\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}} + \frac{6 \log{\left(2 \right)}^{2} \log{\left(\pi - 2 x \right)}^{2}}{\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}} - \frac{4 \log{\left(2 \right)}^{3} \log{\left(\pi - 2 x \right)}}{\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}} + \frac{\log{\left(2 \right)}^{4}}{\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}}\right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{2 x \left(- 3 \tan^{2}{\left(x \right)} - 3\right)}{\tan^{4}{\left(x \right)}} + \frac{2 x \left(- \tan^{2}{\left(x \right)} - 1\right)}{\tan^{2}{\left(x \right)}} - \frac{\pi \left(- 3 \tan^{2}{\left(x \right)} - 3\right)}{\tan^{4}{\left(x \right)}} - \frac{- 2 \tan^{2}{\left(x \right)} - 2}{\tan^{3}{\left(x \right)}} - \frac{\pi \left(- \tan^{2}{\left(x \right)} - 1\right)}{\tan^{2}{\left(x \right)}} + \frac{2}{\tan{\left(x \right)}} + \frac{2}{\tan^{3}{\left(x \right)}}}{\left(- \frac{2 x}{\tan{\left(x \right)}} - \frac{2 x}{\tan^{3}{\left(x \right)}} + 1 + \frac{\pi}{\tan{\left(x \right)}} + \frac{1}{\tan^{2}{\left(x \right)}} + \frac{\pi}{\tan^{3}{\left(x \right)}}\right)^{2} \left(\frac{\left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(\pi - 2 x \right)}^{4}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} - \frac{4 \left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(2 \right)} \log{\left(\pi - 2 x \right)}^{3}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} + \frac{6 \left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(2 \right)}^{2} \log{\left(\pi - 2 x \right)}^{2}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} - \frac{4 \left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(2 \right)}^{3} \log{\left(\pi - 2 x \right)}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} + \frac{\left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(2 \right)}^{4}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} - \frac{8 \log{\left(\pi - 2 x \right)}^{3}}{\left(\pi - 2 x\right) \left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)} + \frac{24 \log{\left(2 \right)} \log{\left(\pi - 2 x \right)}^{2}}{\left(\pi - 2 x\right) \left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)} - \frac{24 \log{\left(2 \right)}^{2} \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right) \left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)} + \frac{8 \log{\left(2 \right)}^{3}}{\left(\pi - 2 x\right) \left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)}\right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{2 x \left(- 3 \tan^{2}{\left(x \right)} - 3\right)}{\tan^{4}{\left(x \right)}} + \frac{2 x \left(- \tan^{2}{\left(x \right)} - 1\right)}{\tan^{2}{\left(x \right)}} - \frac{\pi \left(- 3 \tan^{2}{\left(x \right)} - 3\right)}{\tan^{4}{\left(x \right)}} - \frac{- 2 \tan^{2}{\left(x \right)} - 2}{\tan^{3}{\left(x \right)}} - \frac{\pi \left(- \tan^{2}{\left(x \right)} - 1\right)}{\tan^{2}{\left(x \right)}} + \frac{2}{\tan{\left(x \right)}} + \frac{2}{\tan^{3}{\left(x \right)}}}{\frac{\left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(\pi - 2 x \right)}^{4}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} - \frac{4 \left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(2 \right)} \log{\left(\pi - 2 x \right)}^{3}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} + \frac{6 \left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(2 \right)}^{2} \log{\left(\pi - 2 x \right)}^{2}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} - \frac{4 \left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(2 \right)}^{3} \log{\left(\pi - 2 x \right)}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} + \frac{\left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(2 \right)}^{4}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} - \frac{8 \log{\left(\pi - 2 x \right)}^{3}}{\left(\pi - 2 x\right) \left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)} + \frac{24 \log{\left(2 \right)} \log{\left(\pi - 2 x \right)}^{2}}{\left(\pi - 2 x\right) \left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)} - \frac{24 \log{\left(2 \right)}^{2} \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right) \left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)} + \frac{8 \log{\left(2 \right)}^{3}}{\left(\pi - 2 x\right) \left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{2 x \left(- 3 \tan^{2}{\left(x \right)} - 3\right)}{\tan^{4}{\left(x \right)}} + \frac{2 x \left(- \tan^{2}{\left(x \right)} - 1\right)}{\tan^{2}{\left(x \right)}} - \frac{\pi \left(- 3 \tan^{2}{\left(x \right)} - 3\right)}{\tan^{4}{\left(x \right)}} - \frac{- 2 \tan^{2}{\left(x \right)} - 2}{\tan^{3}{\left(x \right)}} - \frac{\pi \left(- \tan^{2}{\left(x \right)} - 1\right)}{\tan^{2}{\left(x \right)}} + \frac{2}{\tan{\left(x \right)}} + \frac{2}{\tan^{3}{\left(x \right)}}}{\frac{\left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(\pi - 2 x \right)}^{4}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} - \frac{4 \left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(2 \right)} \log{\left(\pi - 2 x \right)}^{3}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} + \frac{6 \left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(2 \right)}^{2} \log{\left(\pi - 2 x \right)}^{2}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} - \frac{4 \left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(2 \right)}^{3} \log{\left(\pi - 2 x \right)}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} + \frac{\left(- \frac{8 \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(\pi - 2 x\right)^{2}} + \frac{8}{\left(\pi - 2 x\right)^{2}}\right) \log{\left(2 \right)}^{4}}{\left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)^{2}} - \frac{8 \log{\left(\pi - 2 x \right)}^{3}}{\left(\pi - 2 x\right) \left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)} + \frac{24 \log{\left(2 \right)} \log{\left(\pi - 2 x \right)}^{2}}{\left(\pi - 2 x\right) \left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)} - \frac{24 \log{\left(2 \right)}^{2} \log{\left(\pi - 2 x \right)}}{\left(\pi - 2 x\right) \left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)} + \frac{8 \log{\left(2 \right)}^{3}}{\left(\pi - 2 x\right) \left(\frac{4 \log{\left(\pi - 2 x \right)}}{\pi - 2 x} - \frac{4 \log{\left(2 \right)}}{\pi - 2 x}\right)}}\right)$$
=
$$\infty$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)
Gráfica
A la izquierda y a la derecha
[src]
/ tan(x) \ lim |-----------| pi | /pi \| x->--+|log|-- - x|| 2 \ \2 //
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\tan{\left(x \right)}}{\log{\left(- x + \frac{\pi}{2} \right)}}\right)$$
oo
$$\infty$$
= (21.619285905718 + 13.5370144752335j)
/ tan(x) \ lim |-----------| pi | /pi \| x->---|log|-- - x|| 2 \ \2 //
$$\lim_{x \to \frac{\pi}{2}^-}\left(\frac{\tan{\left(x \right)}}{\log{\left(- x + \frac{\pi}{2} \right)}}\right)$$
-oo
$$-\infty$$
= -30.095549261586
= -30.095549261586
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to \frac{\pi}{2}^-}\left(\frac{\tan{\left(x \right)}}{\log{\left(- x + \frac{\pi}{2} \right)}}\right) = \infty$$
Más detalles con x→pi/2 a la izquierda
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\tan{\left(x \right)}}{\log{\left(- x + \frac{\pi}{2} \right)}}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{\tan{\left(x \right)}}{\log{\left(- x + \frac{\pi}{2} \right)}}\right)$$
Más detalles con x→oo
$$\lim_{x \to 0^-}\left(\frac{\tan{\left(x \right)}}{\log{\left(- x + \frac{\pi}{2} \right)}}\right) = 0$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{\log{\left(- x + \frac{\pi}{2} \right)}}\right) = 0$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{\tan{\left(x \right)}}{\log{\left(- x + \frac{\pi}{2} \right)}}\right) = \frac{\tan{\left(1 \right)}}{\log{\left(-1 + \frac{\pi}{2} \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\tan{\left(x \right)}}{\log{\left(- x + \frac{\pi}{2} \right)}}\right) = \frac{\tan{\left(1 \right)}}{\log{\left(-1 + \frac{\pi}{2} \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)}}{\log{\left(- x + \frac{\pi}{2} \right)}}\right)$$
Más detalles con x→-oo
Más detalles con x→pi/2 a la izquierda
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\tan{\left(x \right)}}{\log{\left(- x + \frac{\pi}{2} \right)}}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{\tan{\left(x \right)}}{\log{\left(- x + \frac{\pi}{2} \right)}}\right)$$
Más detalles con x→oo
$$\lim_{x \to 0^-}\left(\frac{\tan{\left(x \right)}}{\log{\left(- x + \frac{\pi}{2} \right)}}\right) = 0$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{\log{\left(- x + \frac{\pi}{2} \right)}}\right) = 0$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{\tan{\left(x \right)}}{\log{\left(- x + \frac{\pi}{2} \right)}}\right) = \frac{\tan{\left(1 \right)}}{\log{\left(-1 + \frac{\pi}{2} \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\tan{\left(x \right)}}{\log{\left(- x + \frac{\pi}{2} \right)}}\right) = \frac{\tan{\left(1 \right)}}{\log{\left(-1 + \frac{\pi}{2} \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)}}{\log{\left(- x + \frac{\pi}{2} \right)}}\right)$$
Más detalles con x→-oo
Respuesta numérica
[src]
(21.619285905718 + 13.5370144752335j)
(21.619285905718 + 13.5370144752335j)