x→0时,若x-[sin(x^3)]^1/3与Ax^k是等价无穷小,求A和k

题目

x→0时,若x-[sin(x^3)]^1/3与Ax^k是等价无穷小,求A和k

答案

sin(x^3) = x^3 - x^9/6 + O(x^11)
由[sin(x^3)]^1/3 (x^3)^1/3=x
可设 [sin(x^3)]^1/3 = x +ax^b+o(x^b)
则[x +ax^b+o(x^b)]^3=x^3+3ax^(2+b)+o(x^(b+2))
有 3a=-1/6,2+b=9 即a=-1/18,b=7
∴ x-[sin(x^3)]^1/3 x - ( x - x^7/18x+o(x^7)) = x^7/18