已知函数f(x)在x=0的某邻域内有连续导数,且lim(x→0)[sinx/x^2+f(x)/x]=1,求f‘（0）

Question

已知函数f(x)在x=0的某邻域内有连续导数,且lim(x→0)[sinx/x^2+f(x)/x]=1,求f‘（0）

Answer 1

1=lim(x→0)[sinx/x^2+f(x)/x]
=lim(x→0)[(sinx+xf(x))/x^2] 下面用罗比达法则
=lim(x→0)[(cosx+f(x)+xf'(x))/2x]
=lim(x→0)[(cosx+f(x))/2x]+lim(x→0)f'(x)/2
=lim(x→0)[(-sinx+f'(x))/2]+lim(x→0)f'(x)/2
=f'(0)/2+f'(0)/2
f'(0)=1