limx→0(cosx+xsinx)limx→0(cosx+xsinx)^(1/x^2)

Question

limx→0 (cosx+xsinx)
limx→0 (cosx+xsinx)^(1/x^2)

Answer 1

原式=limx→0（1+xtanx)^(1/x^2)(Cosx)^(1/x^2)
=limx→0(1+x^2)^(1/x^2) (1+Cosx-1)^{[1/(Cosx-1)][(Cosx-1）/x^2]}
=elimx→0e^[(Cosx-1）/x^2]
limx→0e^[(Cosx-1）/x^2]=limx→0(-sinx)/2x=-1/2
所以,原式=exe^(-1/2)=e^(1/2)