若函数f(x)在x=0处连续,且lim{x趋近0}f(x)/x存在,试证f(x)在x=0处可导

Question

若函数f(x)在x=0处连续,且lim{x趋近0}f(x)/x存在,试证f(x)在x=0处可导

Answer 1

证明：
∵limf(x)/x存在,且x→0(当x→0)
∴f(x)→0(当x→0)
又∵f(x)在x=0处连续
∴f(0)=0
limf(x)/x=lim[f(x)-f(0)]/(x-0)=f'(0)
∴f(x)在x=0处可导