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