证 设f(0)=0 U lim F(x)-F(O)=limf()(+ sin-xD-o x→>0x-0 lim f(x)-f(0) (1+| sin x d))=f(0) →>0 故F(x)在x=0处可导 设F(x)在x=0处可导 →lim F(x)-F(0) x→0 x-0 lim/(r)(+ sin x d-f(0) →>0 0证一 设f (0) 0 则 0 ( ) (0) lim 0 x F x F x x f x x x ( )(1 |sin |) 0 lim 0 (1 |sin |) 0 ( ) (0) lim 0 x x f x f x f (0) 故 F(x)在x 0处可导 设F(x)在x 0处可导 0 ( ) (0) lim 0 x F x F x 0 ( )(1 |sin |) (0) lim 0 x f x x f x