分类：教育/科学-升学入学-考研
问题：关于等价无穷小(竞赛题) 
 (题)
回答：1.F(x)=x^2∫{0→x}f(t)dt-∫{0→x}tf(t)dt
==>
F'(x)=[x^2∫{0→x}f(t)dt]'-[∫{0→x}tf(t)dt]'=
=2x∫{0→x}f(t)dt+x^2f(x)-xf(x).

2.Lim{x→0}[2x∫{0→x}f(t)dt+x^2f(x)]/x^2=
=2Lim{x→0}∫{0→x}f(t)dt/x+
+Lim{x→0}f(x)/x*Lim{x→0}x
=2Lim{x→0}[∫{0→x}f(t)dt]'/[x]'+a*0=
=2Lim{x→0}f(x)/[1]=0

Lim{x→0}[xf(x)]/x^2=
Lim{x→0}f(x)/x=a≠0

==>Lim{x→0}[F'(x)]/x^2=a≠0
==>F'(x)和x^2同阶。
n=2