x趋于0，ax(1-cosx)+b(x-sinx)与x^5等价无穷小，求a，b的值？

首先x趋近于0，那么x的五次方首先是趋近于0的，另外看题目的等式。因为x和sinx是等价无穷小，所以x趋近于零，+后面的就等于0。再来看前边的，1-cosx很明显，他是等于2sin²1/2x的，所以a和b的值是1和2

x->0

cosx =1-(1/2)x^2 +(1/24)x^4 +o(x^4)

1-cosx =(1/2)x^2 -(1/24)x^4 +o(x^4)

ax(1-cosx) =(1/2)ax^3 -(1/24)ax^5 +o(x^5)

sinx = x-(1/6)x^3 +(1/120)x^5 +o(x^5)

x-sinx = (1/6)x^3 -(1/120)x^5 +o(x^5)

b(x-sinx) =(1/6)bx^3 -(1/120)bx^5 +o(x^5)

ax(1-cosx) -b(x-sinx) =[(1/2)a -(1/6)b]x^3 +[ -(1/24)a +(1/120)b]x^5 +o(x^5)

ax(1-cosx)+b(x-sinx)与x^5等价无穷小

=>

(1/2)a -(1/6)b =0                                    (1)  and

-(1/24)a +(1/120)b =1                            (2)

20(2) +(1)

(-5/6 + 1/2)a =20

-(1/3)a =20

a=-60

from (1)

(1/2)a -(1/6)b =0

-30 -(1/6)b =0

b= -180

(a,b)=(-60, -180)

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