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Let $x>0$. Assume $f(x)>0$, $f$ is strictly decreasing, and $\lim_{x\downarrow 0} f(x)=+\infty$. If $xf(x)<K<\infty$ for all $x>0$ (where $K$ does not depend on $x$), does the limit $\lim_{x\downarrow 0} xf(x)$ exist? If it does not exists in general, under what conditions does it exist?

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  • $\begingroup$ I have corrected the inconsistency, removing sequences from the title as I want to ask a question about limit of functions. Thanks. $\endgroup$ – Rob Jan 2 '15 at 20:37
  • $\begingroup$ In the problem I am working on, I've just been able to prove that $f(x)$ is strictly decreasing. Can this assumption help? Your example is based on a function $f(x)$ that is not monotone. $\endgroup$ – Rob Jan 2 '15 at 22:32
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A counterexample is given by $$f(x) = \frac{3+\sin \log x}{x}$$ Note that $$f'(x) = \frac{\cos\log x -\sin\log x - 3 }{x^2}<0$$ so $f$ is monotone. Also, $$f''(x) = \frac{6-3\cos\log x + \sin\log x }{x^3}>0$$ so $f$ is convex.

I don't expect there to be a sufficient condition that is easier to verify than the existence of the limit itself.

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