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Límite de la función/ -pi/2/ tan(x)/ tan(x)/log(x-pi/2)

Límite de la función tan(x)/log(x-pi/2)

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Ha introducido [src] 
      /   tan(x)  \
 lim  |-----------|
   pi |   /    pi\|
x->--+|log|x - --||
   2  \   \    2 //
lim⁡x→π2+(tan⁡(x)log⁡(x−π2))\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\tan{\left(x \right)}}{\log{\left(x - \frac{\pi}{2} \right)}}\right) 
Limit(tan(x)/log(x - pi/2), x, pi/2)
Método de l'Hopital
Tenemos la indeterminación de tipo
0/0,

tal que el límite para el numerador es
lim⁡x→π2+1log⁡(2x−π)−log⁡(2)=0\lim_{x \to \frac{\pi}{2}^+} \frac{1}{\log{\left(2 x - \pi \right)} - \log{\left(2 \right)}} = 0
y el límite para el denominador es
lim⁡x→π2+1tan⁡(x)=0\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→π2+(tan⁡(x)log⁡(x−π2))\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→π2+(tan⁡(x)log⁡(2x−π2))\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\tan{\left(x \right)}}{\log{\left(\frac{2 x - \pi}{2} \right)}}\right)
=
lim⁡x→π2+(ddx1log⁡(2x−π)−log⁡(2)ddx1tan⁡(x))\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(2 x - \pi \right)} - \log{\left(2 \right)}}}{\frac{d}{d x} \frac{1}{\tan{\left(x \right)}}}\right)
=
lim⁡x→π2+(1(log⁡(2x−π)2−2log⁡(2)log⁡(2x−π)+log⁡(2)2)(x+xtan⁡2(x)−π2−π2tan⁡2(x)))\lim_{x \to \frac{\pi}{2}^+}\left(\frac{1}{\left(\log{\left(2 x - \pi \right)}^{2} - 2 \log{\left(2 \right)} \log{\left(2 x - \pi \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→π2+(ddx1log⁡(2x−π)2−2log⁡(2)log⁡(2x−π)+log⁡(2)2ddx(x+xtan⁡2(x)−π2−π2tan⁡2(x)))\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(2 x - \pi \right)}^{2} - 2 \log{\left(2 \right)} \log{\left(2 x - \pi \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→π2+(1(log⁡(2x−π)4−4log⁡(2x−π)2x−π+4log⁡(2)2x−π−4log⁡(2)log⁡(2x−π)3−4log⁡(2x−π)2x−π+4log⁡(2)2x−π+6log⁡(2)2log⁡(2x−π)2−4log⁡(2x−π)2x−π+4log⁡(2)2x−π−4log⁡(2)3log⁡(2x−π)−4log⁡(2x−π)2x−π+4log⁡(2)2x−π+log⁡(2)4−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)(−2xtan⁡(x)−2xtan⁡3(x)+1+πtan⁡(x)+1tan⁡2(x)+πtan⁡3(x)))\lim_{x \to \frac{\pi}{2}^+}\left(\frac{1}{\left(\frac{\log{\left(2 x - \pi \right)}^{4}}{- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}} - \frac{4 \log{\left(2 \right)} \log{\left(2 x - \pi \right)}^{3}}{- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}} + \frac{6 \log{\left(2 \right)}^{2} \log{\left(2 x - \pi \right)}^{2}}{- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}} - \frac{4 \log{\left(2 \right)}^{3} \log{\left(2 x - \pi \right)}}{- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}} + \frac{\log{\left(2 \right)}^{4}}{- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}}\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→π2+(ddx1−2xtan⁡(x)−2xtan⁡3(x)+1+πtan⁡(x)+1tan⁡2(x)+πtan⁡3(x)ddx(log⁡(2x−π)4−4log⁡(2x−π)2x−π+4log⁡(2)2x−π−4log⁡(2)log⁡(2x−π)3−4log⁡(2x−π)2x−π+4log⁡(2)2x−π+6log⁡(2)2log⁡(2x−π)2−4log⁡(2x−π)2x−π+4log⁡(2)2x−π−4log⁡(2)3log⁡(2x−π)−4log⁡(2x−π)2x−π+4log⁡(2)2x−π+log⁡(2)4−4log⁡(2x−π)2x−π+4log⁡(2)2x−π))\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(2 x - \pi \right)}^{4}}{- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}} - \frac{4 \log{\left(2 \right)} \log{\left(2 x - \pi \right)}^{3}}{- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}} + \frac{6 \log{\left(2 \right)}^{2} \log{\left(2 x - \pi \right)}^{2}}{- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}} - \frac{4 \log{\left(2 \right)}^{3} \log{\left(2 x - \pi \right)}}{- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}} + \frac{\log{\left(2 \right)}^{4}}{- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}}\right)}\right)
=
lim⁡x→π2+(2x(−3tan⁡2(x)−3)tan⁡4(x)+2x(−tan⁡2(x)−1)tan⁡2(x)−π(−3tan⁡2(x)−3)tan⁡4(x)−−2tan⁡2(x)−2tan⁡3(x)−π(−tan⁡2(x)−1)tan⁡2(x)+2tan⁡(x)+2tan⁡3(x)(−2xtan⁡(x)−2xtan⁡3(x)+1+πtan⁡(x)+1tan⁡2(x)+πtan⁡3(x))2((−8log⁡(2x−π)(2x−π)2+8log⁡(2)(2x−π)2+8(2x−π)2)log⁡(2x−π)4(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)2−4(−8log⁡(2x−π)(2x−π)2+8log⁡(2)(2x−π)2+8(2x−π)2)log⁡(2)log⁡(2x−π)3(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)2+6(−8log⁡(2x−π)(2x−π)2+8log⁡(2)(2x−π)2+8(2x−π)2)log⁡(2)2log⁡(2x−π)2(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)2−4(−8log⁡(2x−π)(2x−π)2+8log⁡(2)(2x−π)2+8(2x−π)2)log⁡(2)3log⁡(2x−π)(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)2+(−8log⁡(2x−π)(2x−π)2+8log⁡(2)(2x−π)2+8(2x−π)2)log⁡(2)4(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)2+8log⁡(2x−π)3(2x−π)(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)−24log⁡(2)log⁡(2x−π)2(2x−π)(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)+24log⁡(2)2log⁡(2x−π)(2x−π)(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)−8log⁡(2)3(2x−π)(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)))\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(2 x - \pi \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8}{\left(2 x - \pi\right)^{2}}\right) \log{\left(2 x - \pi \right)}^{4}}{\left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)^{2}} - \frac{4 \left(- \frac{8 \log{\left(2 x - \pi \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8}{\left(2 x - \pi\right)^{2}}\right) \log{\left(2 \right)} \log{\left(2 x - \pi \right)}^{3}}{\left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)^{2}} + \frac{6 \left(- \frac{8 \log{\left(2 x - \pi \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8}{\left(2 x - \pi\right)^{2}}\right) \log{\left(2 \right)}^{2} \log{\left(2 x - \pi \right)}^{2}}{\left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)^{2}} - \frac{4 \left(- \frac{8 \log{\left(2 x - \pi \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8}{\left(2 x - \pi\right)^{2}}\right) \log{\left(2 \right)}^{3} \log{\left(2 x - \pi \right)}}{\left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)^{2}} + \frac{\left(- \frac{8 \log{\left(2 x - \pi \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8}{\left(2 x - \pi\right)^{2}}\right) \log{\left(2 \right)}^{4}}{\left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)^{2}} + \frac{8 \log{\left(2 x - \pi \right)}^{3}}{\left(2 x - \pi\right) \left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)} - \frac{24 \log{\left(2 \right)} \log{\left(2 x - \pi \right)}^{2}}{\left(2 x - \pi\right) \left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)} + \frac{24 \log{\left(2 \right)}^{2} \log{\left(2 x - \pi \right)}}{\left(2 x - \pi\right) \left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)} - \frac{8 \log{\left(2 \right)}^{3}}{\left(2 x - \pi\right) \left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)}\right)}\right)
=
lim⁡x→π2+(2x(−3tan⁡2(x)−3)tan⁡4(x)+2x(−tan⁡2(x)−1)tan⁡2(x)−π(−3tan⁡2(x)−3)tan⁡4(x)−−2tan⁡2(x)−2tan⁡3(x)−π(−tan⁡2(x)−1)tan⁡2(x)+2tan⁡(x)+2tan⁡3(x)(−8log⁡(2x−π)(2x−π)2+8log⁡(2)(2x−π)2+8(2x−π)2)log⁡(2x−π)4(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)2−4(−8log⁡(2x−π)(2x−π)2+8log⁡(2)(2x−π)2+8(2x−π)2)log⁡(2)log⁡(2x−π)3(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)2+6(−8log⁡(2x−π)(2x−π)2+8log⁡(2)(2x−π)2+8(2x−π)2)log⁡(2)2log⁡(2x−π)2(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)2−4(−8log⁡(2x−π)(2x−π)2+8log⁡(2)(2x−π)2+8(2x−π)2)log⁡(2)3log⁡(2x−π)(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)2+(−8log⁡(2x−π)(2x−π)2+8log⁡(2)(2x−π)2+8(2x−π)2)log⁡(2)4(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)2+8log⁡(2x−π)3(2x−π)(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)−24log⁡(2)log⁡(2x−π)2(2x−π)(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)+24log⁡(2)2log⁡(2x−π)(2x−π)(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)−8log⁡(2)3(2x−π)(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π))\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(2 x - \pi \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8}{\left(2 x - \pi\right)^{2}}\right) \log{\left(2 x - \pi \right)}^{4}}{\left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)^{2}} - \frac{4 \left(- \frac{8 \log{\left(2 x - \pi \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8}{\left(2 x - \pi\right)^{2}}\right) \log{\left(2 \right)} \log{\left(2 x - \pi \right)}^{3}}{\left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)^{2}} + \frac{6 \left(- \frac{8 \log{\left(2 x - \pi \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8}{\left(2 x - \pi\right)^{2}}\right) \log{\left(2 \right)}^{2} \log{\left(2 x - \pi \right)}^{2}}{\left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)^{2}} - \frac{4 \left(- \frac{8 \log{\left(2 x - \pi \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8}{\left(2 x - \pi\right)^{2}}\right) \log{\left(2 \right)}^{3} \log{\left(2 x - \pi \right)}}{\left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)^{2}} + \frac{\left(- \frac{8 \log{\left(2 x - \pi \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8}{\left(2 x - \pi\right)^{2}}\right) \log{\left(2 \right)}^{4}}{\left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)^{2}} + \frac{8 \log{\left(2 x - \pi \right)}^{3}}{\left(2 x - \pi\right) \left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)} - \frac{24 \log{\left(2 \right)} \log{\left(2 x - \pi \right)}^{2}}{\left(2 x - \pi\right) \left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)} + \frac{24 \log{\left(2 \right)}^{2} \log{\left(2 x - \pi \right)}}{\left(2 x - \pi\right) \left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)} - \frac{8 \log{\left(2 \right)}^{3}}{\left(2 x - \pi\right) \left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)}}\right)
=
lim⁡x→π2+(2x(−3tan⁡2(x)−3)tan⁡4(x)+2x(−tan⁡2(x)−1)tan⁡2(x)−π(−3tan⁡2(x)−3)tan⁡4(x)−−2tan⁡2(x)−2tan⁡3(x)−π(−tan⁡2(x)−1)tan⁡2(x)+2tan⁡(x)+2tan⁡3(x)(−8log⁡(2x−π)(2x−π)2+8log⁡(2)(2x−π)2+8(2x−π)2)log⁡(2x−π)4(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)2−4(−8log⁡(2x−π)(2x−π)2+8log⁡(2)(2x−π)2+8(2x−π)2)log⁡(2)log⁡(2x−π)3(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)2+6(−8log⁡(2x−π)(2x−π)2+8log⁡(2)(2x−π)2+8(2x−π)2)log⁡(2)2log⁡(2x−π)2(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)2−4(−8log⁡(2x−π)(2x−π)2+8log⁡(2)(2x−π)2+8(2x−π)2)log⁡(2)3log⁡(2x−π)(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)2+(−8log⁡(2x−π)(2x−π)2+8log⁡(2)(2x−π)2+8(2x−π)2)log⁡(2)4(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)2+8log⁡(2x−π)3(2x−π)(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)−24log⁡(2)log⁡(2x−π)2(2x−π)(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)+24log⁡(2)2log⁡(2x−π)(2x−π)(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π)−8log⁡(2)3(2x−π)(−4log⁡(2x−π)2x−π+4log⁡(2)2x−π))\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(2 x - \pi \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8}{\left(2 x - \pi\right)^{2}}\right) \log{\left(2 x - \pi \right)}^{4}}{\left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)^{2}} - \frac{4 \left(- \frac{8 \log{\left(2 x - \pi \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8}{\left(2 x - \pi\right)^{2}}\right) \log{\left(2 \right)} \log{\left(2 x - \pi \right)}^{3}}{\left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)^{2}} + \frac{6 \left(- \frac{8 \log{\left(2 x - \pi \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8}{\left(2 x - \pi\right)^{2}}\right) \log{\left(2 \right)}^{2} \log{\left(2 x - \pi \right)}^{2}}{\left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)^{2}} - \frac{4 \left(- \frac{8 \log{\left(2 x - \pi \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8}{\left(2 x - \pi\right)^{2}}\right) \log{\left(2 \right)}^{3} \log{\left(2 x - \pi \right)}}{\left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)^{2}} + \frac{\left(- \frac{8 \log{\left(2 x - \pi \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8 \log{\left(2 \right)}}{\left(2 x - \pi\right)^{2}} + \frac{8}{\left(2 x - \pi\right)^{2}}\right) \log{\left(2 \right)}^{4}}{\left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)^{2}} + \frac{8 \log{\left(2 x - \pi \right)}^{3}}{\left(2 x - \pi\right) \left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)} - \frac{24 \log{\left(2 \right)} \log{\left(2 x - \pi \right)}^{2}}{\left(2 x - \pi\right) \left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)} + \frac{24 \log{\left(2 \right)}^{2} \log{\left(2 x - \pi \right)}}{\left(2 x - \pi\right) \left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)} - \frac{8 \log{\left(2 \right)}^{3}}{\left(2 x - \pi\right) \left(- \frac{4 \log{\left(2 x - \pi \right)}}{2 x - \pi} + \frac{4 \log{\left(2 \right)}}{2 x - \pi}\right)}}\right)
=
∞\infty
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)
Gráfica
-3.0-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.53.0-20002000
Trazar el gráfico
Respuesta rápida [src] 
oo
∞\infty
Abrir y simplificar
A la izquierda y a la derecha [src] 
      /   tan(x)  \
 lim  |-----------|
   pi |   /    pi\|
x->--+|log|x - --||
   2  \   \    2 //
lim⁡x→π2+(tan⁡(x)log⁡(x−π2))\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\tan{\left(x \right)}}{\log{\left(x - \frac{\pi}{2} \right)}}\right) 
oo
∞\infty 
= 30.0955492615864
      /   tan(x)  \
 lim  |-----------|
   pi |   /    pi\|
x->---|log|x - --||
   2  \   \    2 //
lim⁡x→π2−(tan⁡(x)log⁡(x−π2))\lim_{x \to \frac{\pi}{2}^-}\left(\frac{\tan{\left(x \right)}}{\log{\left(x - \frac{\pi}{2} \right)}}\right) 
-oo
−∞-\infty 
= (-21.6192859057176 - 13.5370144752333j)
= (-21.6192859057176 - 13.5370144752333j)
Otros límites con x→0, -oo, +oo, 1
lim⁡x→π2−(tan⁡(x)log⁡(x−π2))=∞\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→π2+(tan⁡(x)log⁡(x−π2))=∞\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\tan{\left(x \right)}}{\log{\left(x - \frac{\pi}{2} \right)}}\right) = \infty
lim⁡x→∞(tan⁡(x)log⁡(x−π2))\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→0−(tan⁡(x)log⁡(x−π2))=0\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→0+(tan⁡(x)log⁡(x−π2))=0\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→1−(tan⁡(x)log⁡(x−π2))=tan⁡(1)log⁡(−1+π2)+iπ\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)} + i \pi}
Más detalles con x→1 a la izquierda
lim⁡x→1+(tan⁡(x)log⁡(x−π2))=tan⁡(1)log⁡(−1+π2)+iπ\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)} + i \pi}
Más detalles con x→1 a la derecha
lim⁡x→−∞(tan⁡(x)log⁡(x−π2))\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] 
30.0955492615864
30.0955492615864
    © Sr Examen – Калькулятор