By definition, for a definable functions $f$ on $(0,\mathbb R)$ in a polynomial bounded o-minimal structure, $f(x)=C x^r+o(x^r)$. But I wonder what could happen with the lower order term. Is it possible that $f(x)=c_1x^{a_1}+c_2x^{a_2}+c_3x^{a_3}\cdots$ with endless "powers" ( with real numbers $a_1>a_2>a_3\cdots$)?
My wild guess is that maybe it is possible to have finitely many negative $a_i$'s but there can only be finitely many $a_i$'s can be positive (I mean if one could keep subtracting $c_ix^{a_i}$ from $f(x)$ with $a_i>0$ for infinitely many times). Is this speculation correct? In general, can anyone show me from the axioms of o-minimal structures what we could say about $o(x^r)$?
The reference book I am looking at is Tame Topology and O-minimal Structures by Lou van den Dries.