Initially at least, Hardy fields were defined in terms of germs of real functions at infinity.
[1] Elements of a Hardy field are thus equivalence classes and should be denoted, say, [f]∞ to denote the class of functions that are eventually equal to the representative function f. However, in practice the elements are normally just denoted by the representatives themselves, so instead of [f]∞ one would just write f. If F is a subfield of R then we can consider it as a Hardy field by considering the elements of F as constant functions, that is by considering the number α in F as the constant function fα that maps every x in R to α.
Since the polynomial Q can have only finitely many zeros by the fundamental theorem of algebra, such a rational function will be defined for all sufficiently large x, specifically for all x larger than the largest real root of Q.
Another example is the field of functions that can be expressed using the standard arithmetic operations, exponents, and logarithms, and are well-defined on some interval of the form
In this sense the ordering tells us how quickly all the unbounded functions diverge to infinity.
Indeed, if R is an o-minimal expansion of a field, then the set of unary definable functions in R that are defined for all sufficiently large elements forms a Hardy field denoted H(R).