求极限limx→0[tanx-tan(sinx)]sinx/x^3arctanxlimx→0[tanx-tan(sinx)]sinx/x^3arctanx

Question

求极限limx→0[ tanx-tan(sinx)]sinx/x^3arctanx
limx→0[ tanx-tan(sinx)]sinx/x^3arctanx

Answer 1

原式=limx→0 [tanx-tan(sinx)]/x^3*limx→0 sinx/x*limx→0 x/arctanx
=limx→0 [1/cos^2x-cosx/cos^2(sinx)]/3x^2*1*1
=limx→0 [cos^2(sinx)-cos^3x]/3x^2*limx→0 1/[cos^2x*cos^2(sinx)]
=limx→0 [-2cos(sinx)sin(sinx)cosx+3cos^2xsinx]/6x*1
=limx→0 [3sin2x-2sin(2sinx)]/12x*limx→0 cosx
=limx→0 [6cos2x-4cos(2sinx)*cosx]/12
=(6-4)/12
=1/6.