lim当x趋近于0，（cosx-cos2x）/（1-cosx）的极限

Question

如题所述

Answer 1

lim(x->0) (cosx-cos2x)/(1-cosx) (0/0)
=lim(x->0) (-sinx+2sin2x)/sinx (0/0)
=lim(x->0) (-cosx+4cos2x)/cosx
=(-1+4)/1
=3追问

用无穷小代换定理如何做？多谢，追加悬赏

追答

x->0
cosx ~ 1- (1/2)x^2
1-cosx ~ (1/2)x^2

cosx -cos2x ~ 1- (1/2)x^2 - [1- (1/2)(2x)^2]
= (3/2)x^2
lim(x->0) (cosx-cos2x)/(1-cosx)
=lim(x->0) (3/2)x^2 /[ (1/2)x^2 ]
=3

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