设f(x)在x=0的某一邻域内二阶可导,且lim(x-->0)f(x)/x=0,f'(0)=2.求lim(x-->0)f(x)/x^2

题目

设f(x)在x=0的某一邻域内二阶可导,且lim(x-->0)f(x)/x=0,f''(0)=2.求lim(x-->0)f(x)/x^2
因为f(x)在x=0处二阶可导从而连续且lim(x-->0)f(x)/x=0
为什么能得到lim(x-->0)f(x)=f(0)=0.
请详细解释,多谢

答案

因f(x)在x=0处二阶可导从而连续f'(x)=lim(x-->0){[f(x)-f(0)]/x} =lim(x-->0) {-f(0)/x},x-->0,f'(x) 有意义(二阶可导从而连续),除非f(0)=0 (分母x趋于0,则分子必趋于0)lim(x-->0) f(x)/x^2=lim(x-->0)f'(x)/(2x) (...