Question

下图中式子为什么成立？

条件是x趋近于0

Answer 1

x无限趋近于零，左侧1-³√1=0（即左侧无限趋近于0），右侧1/3（1-1）=0（即右侧无限趋近于0）。所以两者相似

Answer 2

x->0
cos2x-1 ->0
[1+(cos2x-1)]^(1/3) ~ 1 +(1/3)(cos2x-1)
1-[1+(cos2x-1)]^(1/3) ~ -(1/3)(cos2x-1) =(1/3)(1-cos2x)

追问

[1+(cos2x-1)]^(1/3) ~ 1 +(1/3)(cos2x-1)，请问这一行是怎么实现的呢？

追答

u->0
(1+u)^(1/3) = 1+(1/3)u +o(u)
x->0, (cos2x-1) ->0

带入
u=(cos2x-1)
[1+(cos2x-1)]^(1/3) ~ 1 +(1/3)(cos2x-1)

追问

u->0
(1+u)^(1/3) = 1+(1/3)u +o(u)
x->0, (cos2x-1) ->0
请问这个式子来自哪里呢？

u->0
(1+u)^(1/3) = 1+(1/3)u +o(u)
x->0, (cos2x-1) ->0
请问这个式子来自哪里呢？

追答

lim(x->0) (cos2x-1)
=1-1
=0
x->0, (cos2x-1) ->0