The term limit comes about relative to a number of topics from several different branches of mathematics.
A sequence 
of elements in a topological space 
is said to have limit 
provided that for each neighborhood 
of 
, there exists a natural number 
so that 
for all 
. This very general definition can be specialized in
the event that 
is a metric space, whence one says that a sequence 
in 
has limit 
if for all 
, there exists a natural
number 
so that
 |
(1)
|
where 
is the metric,
and the inequality holds for
all 
. In many commonly-encountered
scenarios, limits are unique, whereby one says that 
is the limit of 
and writes
 |
(2)
|
Limits in a metric space are unique. On the other hand, a sequence in a topological
space 
may have several, even infinitely many,
different limits provided that 
is equipped with a topology which fails to be T2.
One reads the expression in (2) as "the limit as 
approaches infinity of 
is 
."
The topological notion of convergence can be rewritten to accommodate a wider array of topological spaces 
by utilizing the language of nets.
In particular, if 
is a net from a directed set 
into 
, then an element 
is said to be the limit of 
iff for every neighborhood 
of 
, 
is eventually in 
,
i.e., if there exists an 
so that, for every 
with 
, the point 
lies in 
. This notion is particularly well-purposed for topological
spaces which aren't first-countable.
A function 
is said to have a finite limit 
if, for all 
, there exists a 
such that 
whenever 
. This form of definition is sometimes called
an epsilon-delta definition. This can
be adapted to the case of infinite limits as well: The
limit of 
as 
approaches 
is equal to 
(respectively 
) if for every number 
(respectively 
), there exists a number 
depending on 
for which 
(respectively, 
) whenever 
.
Limits may be taken from below
 |
(3)
|
or from above
 |
(4)
|
For a real variable 
, if the two are equal, then "the"
limit is said to exist
 |
(5)
|
The expression in (3) is read "the limit as 
approaches 
from the left / from below" or "the limit as 
increases to 
," while (4) is read "the limit as 
approaches 
from the right / from above" or "the limit as 
decreases to 
." In (5), one simply refers to "the limit as 
approaches 
."
For a real function 
defined for all sufficiently large real 
, a finite real
number 
is its limit at 
iff
 |
(6)
|
where 
and 
range over the real numbers
and 
is chosen so that 
is defined for every 
. For a limit at 
, replace 
by 
and require 
to be defined for every 
.
For a function of two real variables, a limit must be independent of the path of approach. For
example, the function 
satisfies the inequality
 |
(7)
|
for 
. Since 
tends to 0 as 
approaches the origin, the
squeeze theorem gives
 |
(8)
|
In contrast, the function 
has different limits along the lines 
and 
:
 |  |  |
(9)
|
 |  |  |
(10)
|
Consequently, its limit at the origin does not exist. The relation 
also shows that agreement along both coordinate axes is
insufficient to establish a limit.
Limits are implemented in the Wolfram Language as Limit[f,
x-> x0]. This command also takes the option Direction,
which specifies the direction of approach and can be set to 
, 
,
I, or -I, among other settings.
Note that the function definition of limit can be thought of as a natural generalization of the sequence definition
due to the fact that a sequence 
in a topological
space 
is nothing more than a function 
mapping 
to 
.
For a sequence 
of real numbers, its lower
limit 
and upper limit 
are defined in the affinely
extended real numbers by the limits of the tail infima
and suprema:
 |
(11)
|
 |
(12)
|
These values always exist in the affinely extended real numbers and may equal 
or 
.
For a bounded sequence, they are the least and
greatest subsequential limits. Thus, in the bounded case, 
is the least real number for
which every 
has 
for infinitely many
, and 
is the greatest such real number.
Related notions include supremum limit and infimum limit.
Indeterminate forms of types 
and 
can often be computed with L'Hospital's
rule. Types 
can be converted to the form 
by writing
 |
(13)
|
For 
, types 
, 
,
and 
are treated by introducing a dependent variable
 |
(14)
|
so that
![lny=g(x)ln[f(x)],](https://mathworld.wolfram.com/images/equations/Limit/NumberedEquation13.svg) |
(15)
|
then calculating 
.
The original limit then equals 
,
 |
(16)
|
The indeterminate form 
is also frequently encountered.
All of the above notions can be generalized even further by utilizing the language of ultrafilters. In particular, if 
is a topological space
and if 
is an ultrafilter
on 
, then an element 
is said to be a limit of 
if every neighborhood of
belongs to 
. Several authors have defined similar ideas relative to filters as well (Stadler and Stadler 2002).
The June 2, 1996 comic strip FoxTrot by Bill Amend (Amend 1998, p. 19; Mitchell 2006/2007) featured the following limit as a "hard" exam problem intended for a remedial math class but accidentally handed out to the normal class:
 |
(17)
|
Reproduced with permission of the author.
-
Definition of Limit with Mathematica."
Mathematica Educ. 3, 17-21, 1994.Smith, W. K. 