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Let f be a function that has a finite limit at infinity. It is true that this alone is not enough to show that its derivative converges to zero at infinity. So I was wondering weather there were any additional conditions for f that could give the desired outcome. I am also aware of Barbalat's Lemma but this requires uniform continuity, a property which in many occasions is not easy to verify. Thank you

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If its derivative's limit exists, then it should be zero. – Eastsun Nov 11 '12 at 10:40
What Eastsun says sounds right. For an example of a function with limit 0, but the derivative lacks a limit, see $\sin(x^2)/x.$ – Per Alexandersson Nov 11 '12 at 10:41

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