Question

lim x->inf (2/pi*arctg(x))^x

Answer 1

l = lim x->ꝏ (2/pi • arctg(x))^x =

= lim x->ꝏ (1+2/pi • arctg(x)-1)^x =

= lim x->ꝏ (1+(2arctg(x)-pi)/pi)^x =

= lim x->ꝏ (1+(2arctg(x)-pi)/pi)^(pi/(2arctg(x)-pi)•x•(2arctg(x)-pi)/pi) =

= e^(lim x->ꝏ (x•(2arctg(x)-pi)/pi))

lim x->ꝏ (x•(2arctg(x)-pi)/pi) =

= lim x->ꝏ (2arctg(x)-pi)/(pi/x) = {0/0} =

= lim x->ꝏ (2/(1+x²))/(-pi/x²) =

= lim x->ꝏ (2x²/(-pix²-pi) = -2/pi

=> l = e^(-2/pi)