F(X)在X点处二阶可导，求LIM[F(X+H)-2F(X)+F(X-H)]/H^2 H趋于0

来源：百度知道 编辑：UC知道 时间：2024/08/28 23:47:49
标答是 F''(X)

以下极限都是h趋于0

lim [f(x+h)-2f(x)+f(x-h)]/h^2
(用洛必塔法则)
=lim [f'(x+h)-2f'(x)+f'(x-h)]/2h

=lim (1/2){[f'(x+h)-f'(x)]/h-[f'(x-h)-f'(x)]/(-h)}
=(1/2)[f''(x)-f''(x)]
=0

注意：
lim [f(x+h)-f(x)]/h=lim [f(x-h)-f(x)]/(-h)=f'(x)

等于F''(X)

lim [f(x+h)-2f(x)+f(x-h)]/h^2
(用洛必塔法则)
=lim [f'(x+h)-2f'(x)+f'(x-h)]/2h

=lim (1/2){[f'(x+h)-f'(x)]/h-[f'(x-h)-f'(x)]/(-h)}
=(1/2)[f''(x)-f''(x)]
=0