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A quick check: Am I right in thinking that "$f(x)$ is a slowly varying function wrt $x$" translates to $f^n(x)$ small for all $n\geq1$? Thank you.

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Thank you, Jonas. I need to Taylor expand this "slowly varying function" but not only at $x\to\infty$. Is there a more "applied math" definition? – hil Aug 27 '11 at 21:07
You could explain how you define applied math definition. – Did Aug 27 '11 at 22:00
@Didier: I suppose one that allows me to apply to a Taylor expansion, such that I can determine what order terms to ignore...? Sorry, I know this is a bit vague :-) – hil Aug 27 '11 at 22:09
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The quest for such conditions sounds very much pure math to me, but nevermind. // Re your question, slowly varying functions can behave quite wildly with respect to differentiation, consider for example the first derivative $u'$ when $u(x)=\log(x)+\sin(x^2)$. – Did Aug 27 '11 at 22:21
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