Question

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

Answer 1

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) (洛毕达法则)
=lim(x-->0)f"(x)/2=2/2=1

Answer 2

lim(x-->0)f(x)/x=0，说明f(x)与x比较是一个高阶无穷小，

Answer 3

lim(x-->0)f(x)/x=0
分子x趋于0，则分母必趋于0，否则该极限lim(x-->0)f(x)/x!=0
lim(x-->0)f(x)/x=lim(x-->0)[f(x)-f(0)]/(x-0)=f'(0)=0

由泰勒公式f(x)=f(0)+f'(0)x+(1/2)f"(0)x^2+o(x^2)
则lim(x-->0)f(x)/x^2
=lim(x-->0)[f(0)+f'(0)x+(1/2)f"(0)x^2+o(x^2)]/x^2=lim(x-->0)[(1/2)f"(0)x^2+o(x^2)]/x^2=f"(0)/2 =1

Answer 4

因为f(x)在x=0处二阶可导从而连续且lim(x-->0)f(x)/x=0