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For any function in the Schwartz space $\phi\in {\mathcal{S}({\bf R})}$, what can one say for the following two limits? $$\begin{align} 1.& \qquad\lim_{x\to +\infty}\left(\phi(-x)-\phi(x)\right)[\log x]\\ 2.& \qquad\lim_{x\to 0^+}\left(\phi(-x)-\phi(x)\right)[\log x] \end{align} $$

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up vote 6 down vote accepted

(I hope I understand what you mean) $\phi$ decreases at $\infty$ faster than any power - so the 1st limit is surely $0$. It is also (even infinitely) differentiable, in particular $\lim\limits_{x\to 0} (\phi(x)-\phi(-x))/x$ exists, so the 2nd limit is $0$ too.

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@Theo&Jack: I have to say I'm confused by the notation - what is inside the log? Only $x$ (as I supposed), or everything? –  user8268 May 1 '11 at 21:09
    
I think now the formula is clear. –  Jack May 1 '11 at 21:19
    
Yes, it is and now I understand the answer as well. –  t.b. May 1 '11 at 21:20

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