设函数y=f(x)在（0,+无穷）内有界且可导,则（B)A.当(x→+无穷）limf(x)=0时,必有(x→+无穷）limf'(x)=0.B.当(x→+无穷）limf'(x)存在时,必有(x→+无穷）limf'(x)=0.C.当（x→0^+)limf(x)=0时,必

Question

设函数y=f(x) 在（0,+无穷）内有界且可导,则（B )
A.当( x →+无穷）lim f(x) =0 时,必有( x →+无穷）lim f'(x) =0 .
B.当( x →+无穷）lim f'(x)存在时,必有( x →+无穷）lim f'(x) =0 .
C.当（x→0^+) lim f(x) =0时,必有（x→0^+) lim f'(x) =0.
D.当（x→0^+) lim f'(x)存在时,必有（x→0^+) lim f'(x) =0.（怎么判断）

Answer 1

lim (f(2x)-f(x))/x = lim xf'(c)/x= lim f'(c)
c在x和2x之间
由于f'(x)极限存在
所以lim f'(x) = lim f'(c) = 0
选B