Question

【高数】高阶无穷小的一道题我这样做为什么不对

“欲让上式结果为非零常数，只需分子分母齐次，因为分子是一次，所以分母也为一次，所以k-2=1”这样子做为什么不正确？

Answer 1

f(x) =x^3-3x+2 =>f(1)= 0
f'(x) =3x^2-3 =>f'(1)/1! = 0
f''(x) =6x => f''(1)/2!= 3
f'''(x) =6 => f'''(1)/3!= 1
f(x)
=f(1) +[f'(1)/1!](x-1)+[f''(2)/1!](x-1)^2 +[f'''(1)/3!](x-1)^3
=0+0 +3(x-1)^2 +(x-1)^3
=3(x-1)^2 +(x-1)^3
f(x) 与 (x-1)^k 同阶无穷小
=> k=2

Answer 2

约分之後分子还是无穷小吗???