Question

求极限：x→0时，(x-sinxcosx)/x³

Answer 1

x->0

sin2x = 2x -(1/6)(2x)^3 +o(x^3) = 2x- (4/3)x^3 +o(x^3)

x-(1/2)sin2x = (2/3)x^3 +o(x^3)

lim(x->0) ( x- sinx.cosx) /x^3

=lim(x->0) [ x-  (1/2)sin2x ] /x^3

=lim(x->0) (2/3)x^3  /x^3

=2/3

追问

为什么不能先把cosx消去，用x-sinx算？
对于另一个函数：(x+sinxcosx)/x，不论是否把cosx消去，答案都是2.

追答

当然不可以！
lim(x->0) ( x- sinx.cosx) /x^3
=lim(x->0) [ x- (1/2)sin2x ] /x^3 (0/0 分子分母分别求导)
=lim(x->0) -cos2x /(3x^2) (0/0 分子分母分别求导)
=lim(x->0) 2sin2x/(6x)
=lim(x->0) 2(2x)/(6x)
=2/3