设f(x)=x^n•sin(1/x)(x≠0),且f(0)=0,则f(x)在x=0处()设f(x)=x^n•sin(1/x)(x≠0),且f(0)=0,则f(x)在x=0处()A.\x05仅当lim(x--0)f(x)=lim(x--0)x^n•sin(1/x)=f(0)=0时,才可微B.\x05在任何条件都可微C.\x05当且仅当

Question

设f(x)=x^n•sin(1/x) (x≠0),且f(0)=0,则f(x)在x=0处( )
设f(x)=x^n•sin(1/x) (x≠0),且f(0)=0,则f(x)在x=0处( )
A.\x05仅当lim(x--0)f(x)=lim(x--0) x^n•sin(1/x)=f(0)=0时,才可微
B.\x05在任何条件都可微
C.\x05当且仅当n>1时才可微
D.\x05因sin(1/x)在x=0处无定义,所以不可微

Answer 1

n>1时有
f'(0)=lim (f(x)-f(0))/x = lim x^(n-1)sin(1/x) =0 (无穷小乘有界量)
n