lim(x→0)tanx-sinx/x3方求极限

Question

lim(x→0) tanx-sinx/x3方
求极限

Answer 1

lim(x→0) (tanx-sinx)/x³
=lim(x→0) (sinx/cosx-sinx)/x³
=lim(x→0) (sinx-sinxcosx)/(x³cosx)
=lim(x→0) sinx(1-cosx)/(x³cosx)
=lim(x→0) sinx(1-cosx)/(x³cosx)*sin²x/sin²x
=lim(x→0) sin³x/x³*(1-cosx)/(sin²xcosx)
=lim(x→0) (sinx/x)³*(1-cosx)/[(1-cos²x)cosx]
=lim(x→0) (sinx/x)³*(1-cosx)/[(1+cosx)(1-cosx)cosx]
=lim(x→0) (sinx/x)³*1/[(1+cosx)cosx]
=(1)³*1/[(1+1)*1],定理lim(x→0) (sinx/x)^n=1^n=1
=1/2