Informally, the term asymptotic means approaching a value or curve arbitrarily closely (i.e., as some sort of limit is taken). A line
or curve 
that is asymptotic to given curve 
is called the asymptote
of 
.
More formally, let 
be a continuous variable tending to some limit. Then a real function 
and positive function 
are said to be asymptotically equivalent, written 
, if
 |
(1)
|
as the limit is taken.
Equivalently, consider the little-o asymptotic notation 
that is one of the Landau symbols. Then 
means that
 |
(2)
|
as a limit is taken. The statement 
is then equivalent to
 |
(3)
|
or
 |
(4)
|
(Hardy and Wright 1979, pp. 7-8).
These definitions can also be applied to the discrete case of 
an integer variable that tends to infinity, 
a real function of 
, and 
a positive function of 
.
