lim-x-3-pi-arc-tan-x-2x-

Question Number 83919 by john santu last updated on 08/Mar/20

\lim_{x \to + \infty} {(\frac{3}{π} arc \tan x)}^{2 x} = ?

Commented by john santu last updated on 08/Mar/20

= e^{\lim_{x \to + \infty} (\ln {(\frac{3}{π} \tan^{- 1} (x))}^{2 x})}

= e^{\lim_{x \to + \infty} (\frac{\ln (\frac{3}{π} \tan^{- 1} (x))}{\frac{1}{2 x}})}

= e^{\lim_{x \to + \infty} \frac{π}{{3 \tan}^{- 1} (x)} . \frac{3}{π (1 + x^{2})} . (- 2 x^{2})}

= e^{\lim_{x \to + \infty} \frac{π}{3 (\frac{π}{2})} \times \lim_{x \to + \infty} \frac{- 6 x^{2}}{π + π x^{2}}}

= e^{\frac{2}{3} \times (- \frac{6}{π})} = e^{- \frac{4}{π}}

= \frac{1}{\sqrt[π]{e^{4}}}