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If $F(t)$ is twice differentiable at $x$ and $$G(h)=\max_{t\in(0,h)}\left[\frac{F'(x+t)-F'(x-t)}{2t}-F''(x)\right],$$ where $x$ is fixed; then how can we show that $\displaystyle\lim_{h\to 0}G(h)=0$.

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There are abstract duplicates of this question here and here. –  Zev Chonoles Jun 9 '12 at 15:48
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4 Answers

up vote 1 down vote accepted

Hint: $$\frac{F'(x+t)-F'(x-t)}{t}=\frac{F'(x+t)-F'(x)}{t}+\frac{F'(x-t)-F'(x)}{-t}.$$

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@Thomas: Indeed very useful! I started with the obvious one, then tried to give a little more. –  André Nicolas Jun 9 '12 at 15:57
    
You removed the $t$, I removed my comment. A pity, this equation would have allowed me to show (no, I won't tell ;-) –  user20266 Jun 9 '12 at 15:59
    
@Kns: I was being too indirect. Have changed the hint, added primes. Note that each part is related to the derivative of $F'$. –  André Nicolas Jun 9 '12 at 16:05
    
Thanks a lot Andre! –  Kns Jun 9 '12 at 16:17
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Hint: By definition

$$ F''(x) = \lim_{h\to 0} \frac{F'(x+h)-F'(x)}{h}$$

Couple this with Andre's comment

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Since $\,t\in(,h)\,$ , we have that $\,h\to 0\Longrightarrow t\to 0\,$ , so: $$\lim_{t\to 0}\frac{F'(x+t)-F'(x-t)}{2t}=\lim_{t\to 0}\frac{1}{2}\left[\frac{F'(x+t)-F'(x)}{t}+\frac{F'(x-t)-F'(x)}{-t}\right]$$ and you get what you want since we know $\,F''(x)\,$ exists, so the limit defining this second derivative exists.

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Since $h\to0$ means $t\to0$, so $$\displaystyle\lim_{h\to 0}G(h) = \lim_{t\to 0}\left[\frac{F'(x+t)-F'(x-t)}{2t}-F''(x)\right]$$ apply l'Hôpital's rule,we can get $$ \lim_{t\to 0}\left[\frac{F'(x+t)-F'(x-t)}{2t}-F''(x)\right] = \lim_{t \to 0}\left[\frac{F''(x+t)+F''(x-t)}{2}-F''(x)\right] = 0$$

ps: your tags include real analysis, I assume $x\in \mathbf{R^{n}}$, although real analysis isn't only about real numbers. I dou't know if l'Hôpital's rule can apply under other situation.

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(This was my first intent, too: L'Hospital) I've a doubt here: in the RHS the $\,t's\,$ disappeared while you still are taking the limit when $\,t\to 0\,$ , but this is equivalent to plug $\,t=0\,$ after applying L'Hospital, which is justified if $\,F''\,$ is continuous in $\,x\,$, something we can't know... –  DonAntonio Jun 9 '12 at 16:27
    
If I recall correctly, L'Hospital does not require the derivative to be continuous(namely $F''$ here), the existent of the derivative in the area is enough –  haohaolee Jun 9 '12 at 17:05
    
Indeed it doesn't, but you require it to put $$\lim_{t\to 0}\frac{F''(x+t)+F''(x-t)}{2}=\frac{F''(x)+F''(x)}{2}$$which is what in fact you did, or, of course, justify otherwise this equality. –  DonAntonio Jun 9 '12 at 17:13
    
oh, you are right, my fault, thanks. edit it. And then I get what you meant.... thinking now –  haohaolee Jun 9 '12 at 17:18
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seems it is not appropriate here. Thanks for the reminding –  haohaolee Jun 9 '12 at 17:28
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