设函数f(x)在x=0点连续 且满足limx->0(sinx/x^2+f(x)/x)=2求f'(0)

Question

设函数f(x)在x=0点连续 且满足limx->0(sinx/x^2+f(x)/x)=2求f'(0)

Answer 1

简单分析一下，答案如图所示

Answer 2

∵limx->0(sinx/x^2+f(x)/x)

=limx->0[sinx+xf(x)]/x^2

=limx->0[cosx+f(x)+xf'(x)]/(2x)
=1/2limx->0[cosx+f(x)+xf'(x)]/x
=2
limx->0[cosx+f(x)+xf'(x)=0

limx->0f(x)=-1
limx->0[cosx+f(x)]/x
=limx->0[-sinx+f'(x)]=f'(x)
∴limx->0[cosx+f(x)+xf'(x)]/x
∴limx->0[cosx+f(x)]/x+limx-->0f'(x)

=2f'(0)
=4

∴f'(0)=2本回答被提问者和网友采纳