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Prove that if $\lim\limits_{x\to\infty}f(x)$ and \lim\limits_{x\to\infty}f''(x) exist, then \lim\limits_{x\to\infty}f'(x)=0.

I can prove that \lim\limits_{x\to\infty}f''(x)=0. Otherwise f'(x) goes to infinity and $f(x)$ goes to infinity, contradicting the fact that $\lim\limits_{x\to\infty}f(x)$ exists. I can also prove that if \lim\limits_{x\to\infty}f'(x) exists, it must be 0. So it remains to prove that \lim\limits_{x\to\infty}f'(x) exists. I'm stuck at this point.

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    These questions are two parts of a single question in Spivak's text.2011-11-07

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This is similar to a recent Putnam problem, actually. By Taylor's theorem with error term, we know that for any $x$, f(x+1) = f(x) + f'(x) + \tfrac12f''(t) for some $x\le t\le x+1$. Solve for f'(x) and take limits....

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    Nice! But there should be a solution without Taylor's theorem, because it has not been introduced in this part of the book.2011-11-07
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Hint $\ $ This follows easily from L'Hôpital's rule since

\rm \lim_{x\to\infty}\ f-f'\ =\ \lim_{x\to\infty}\frac{e^x\ (f-f')}{e^x}\ =\ \lim_{x\to\infty}\frac{e^x\ (\:f-f'+f'-f'')}{e^x}\ =\ \lim_{x\to\infty}\ f-f''\ exists

See also the similar classic Hardy old problem.

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    Very clever one!2012-04-11