Definition of "asymptotic"

F. Skof investigated an interesting asymptotic property of the additive functions (see Theorem 2.34). In fact, she proved that a function f : E1 → E2 is additive if and only if ‖f(x + y) − f(x) − f(y)‖ → 0 as ‖x‖ + ‖y‖ → ∞, where E1 is a normed space and E2 is a Banach space.

2011, Soon-Mo Jung, Hyers–Ulam–Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, page 130

More recently, a direction of research initiated by Macpherson and Steinhorn and continued by Elwes [13, 14] and Ryten studies classes of finite structures in which definable sets have a uniform asymptotic behaviour, as the cardinalities of the universes increase.

2011, Vera Koponen, "Some connections between finite, infinite model theory", Finite and Algorithmic Model Theory, Cambridge University Press, page 110