求limx→+∞（2πarctanx）x．

Question

求

lim

x→+∞

（(2)/(π)arctanx）^x．

Answer 1

由于（(2)/(π)arctanx）^x=exln(2)/(π)arctanx，而

lim

x→+∞

xln(2)/(π)arctanx=

lim

x→+∞

(ln2-lnπ+lnarctanx)/((1)/(x))
=

lim

x→+∞

((1)/(arctanx)•(1)/(1+x2))/(-(1)/(x2))=-

lim

x→+∞

(x2)/(arctanx(1+x2))=-(2)/(π)
∴原式=e-(2)/(π)