证明:设f(x)在x=0连续,且lim(x→0) (f(x)/x)=1,则必有f'(0)=1
问题描述:
证明:设f(x)在x=0连续,且lim(x→0) (f(x)/x)=1,则必有f'(0)=1
答
由lim(x→0)[f(x)/x]=1可知f(0)=0
f'(0)=lim(x→0){[f(x)-f(0)]/(x-0)}=lim(x→0)[f(x)/x]=1
答
因为lim(x→0) (f(x)/x)=1 所以,x与f(x)为等价无穷小:f(x) .x趋于0时,f(x)也趋于0
所以:f(0)=0
f'(0)= lim(x→0) [f(x)-f(0)]/(x-0)
= lim(x→0) f(x)/x
= 1