求x趋向于正无穷大时(2/兀arctanx)的x次方的极限

Question

求x趋向于正无穷大时(2/兀arctanx)的x次方的极限

Answer 1

lim(x→∞) (2/π*arctanx)^x
=e^lim(x→∞) xln(2/π*arctanx)
=e^lim(x→∞) ln(2/π*arctanx)/(1/x)
用洛必达法则得
=e^lim(x→∞) 1/[(x^2+1)arctanx]/(-1/x^2)
=e^-lim(x→∞) x^2/[(x^2+1)arctanx]
=e^-lim(x→∞) x^2/(x^2*arctanx+arctanx)
=e^-lim(x→∞) 1/[arctanx+(arctanx)/x^2],取得极限
=e^-1/(π/2+0)
=e^(-2/π)