设f(x)在x=0处可导,则lim(h趋于0)(f(3h)-f(-h))/2h=?
问题描述:
设f(x)在x=0处可导,则lim(h趋于0)(f(3h)-f(-h))/2h=?
答
因为f(x)在0处可导,则f'(0)存在
那么:h→0:lim[f(3h)-f(-h)]/2h=2*lim[f(0+3h)-f(0-h)]/4h=2f'(0)
有不懂欢迎追问
答
[f(0+3h)-f(0)+f(0)-f(0-h)]/2h
=[f(0+3h)-f(0)]/2h+[(f(0)-f(0-h)]/2h
由导数定义可知lim(h趋于0)[f(0+3h)-f(0)]/2h=(3/2)f'(0)
lim(h趋于0)[(f(0)-f(0-h)]/2h=(1/2)f'(0)
于是原式=2f'(0)