一道求极限的问题当x趋近于正无穷,求[(2/∏)*arctanx]^x

Question

一道求极限的问题
当x趋近于正无穷,求[(2/∏)*arctanx]^x

Answer 1

lim x->infinity [(2/π)*arctanx]^x y=arctanx x=tany
= lim y-> π/2 [(2/π)*y]^tany y=π/2-z
= lim z->0 [(2/π)*(π/2-z)]^tan(π/2-z)
tan(π/2-z)=cotz=1/tanz
= lim z->0 [1-2z/π]^(1/tanz) z->0,tanz~z
= lim z->0 [1-2z/π]^(1/z)
= lim z->0 [1-2z/π]^[(1/z)*(-π/2)*(-2/π)]
(1+m)^(1/m)=e m->infinity or 0
=e^(-2/π)