Exponentiation is Hard to Avoid

Let <!-- MATH $\mathcal{R}$ --> be an O-minimal expansion of the field of real numbers. If <!-- MATH $\mathcal{R}$ --> is not polynomially bounded, then the exponential function is definable (without parameters) in <!-- MATH $\mathcal{R}$ --> . If <!-- MATH $\mathcal{R}$ --> is polynomially bounded, then for every definable function <!-- MATH $f:\mathbb{R} \to \mathbb{R}$ --> , f not ultimately identically 0, there exist c, <!-- MATH $r \in \mathbb{R},c \ne 0$ --> , such that <!-- MATH $x \mapsto {x^r}:(0, + \infty ) \to \mathbb{R}$ --> is definable in <!-- MATH $\mathcal{R}$ --> and <!-- MATH ${\lim _{x \to + \infty }}f(x)/{x^r} = c$ --> .