f(x)在x=0的領域內有二階導數,又x→0時lim((sinx+xf(x))\x3)=1/2,求f(0),f'(0),f''(0)

f(x)在x=0的領域內有二階導數,又x→0時lim((sinx+xf(x))\x3)=1/2,求f(0),f'(0),f''(0)

根據洛筆答法則,lim((sinx+xf(x))/x3)=lim((cosx+f(x)+x·f'(x))/3x²;)若x→0時這個極限存在,則必有lim cosx+f(x)+x·f'(x)=0則cos0+f(0)=0f(0)=-1再進一步用洛筆答法則得lim((cosx+f(x)+x·f'(x))/3x²;)=li…