求极限limx→0(tanx-sinx)/x3．
limx→0(tanx-sinx)/x3=limx→0(sinx/cosx)-sinx/x3 =limx→0[sinx(1-cosx)/cosx]/x3=limx→0sinx(1-cosx)/x3cosx =limx→0(sinx/x)•(1/cosx)•[(1-cosx)/x2] 因为limx→0(sinx/x)=1，limx→0(1/cosx)=1 而limx→0(1-cosx)x2=limx→02sin2(x/2)/x2=(1/2)limx→0sin2(x/2)/(x/2)2=1/2 所以limx→0[(tanx-sinx)/x3]=1/2
