令An=f(x),Bn=g(x) an=An-A,bn=Bn-B.
则liman=lim(An-A)
=limAn-lim(-A)
=A-A
=0.
同理limbn=O.
.·. lim(An· Bn)
=lim[(an+ A)(bn+ B)]=lim(an·bn+B·an+A·bn+ AB)=lim(an· bn)+ lim(B*an)+lim(A· bn)+limAB
=0+ B· liman+A· limbn+limAB
=BxO+AxO+AB
=AB=limf(x)*limg(x)
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