Example 2

A very important example of a Hardy field is the set of rational functions with real coefficients, which we shall denote by $\mathbf{R}(x)$, by an abuse of language; if $f(x) = p(x)/q(x)$ is a rational function with real coefficients, not identically zero, it is continuous, differentiable, and $\neq 0$ on an interval $[a, +\infty[$, where $a$ is strictly greater than the largest of the real roots of the polynomials $p(x)$ and $q(x)$; thus every element of $\mathbf{R}(x)/R_\infty$ other than 0 is invertible. We note again that two rational functions cannot be congruent modulo $\mathbf{R}_\infty$ unless they are equal, so $\mathbf{R}(x)/\mathbf{R}_\infty$ can again be identified with $\mathbf{R}(x)$.