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
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