若函数f(x)在x=0处连续且limx→0f(x)/x存在,试证f(x)在x=0处可导

Question

如题所述

Answer 1

简单分析一下，详情如图所示

Answer 2

因为 f(x)在x=0处连续且limx→0 f(x)/x 存在
所以 f(0) = lim (x-->0) f(x)
= lim (x-->0) f(x)/x * x = lim (x-->0) f(x)/x * lim (x-->0) x = 0
于是：设 limx→0 f(x)/x = A
lim (x-->0) |（f(x) - f(0)) / (x -0) - A| = lim (x-->0) |f(x) / x - A| = | lim (x-->0) f(x) / x - A | = 0
即 f'(0) = A 存在