Limit
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 ![<span style=]()
{x_n}" src="http://mathworld.wolfram.com/images/equations/Limit/Inline10.gif" style="height:14px; width:23px" /> in 
has limit 
if for all 
, there exists a natural number 
so that
 |
(1)
|
for all 
. In many commonly-encountered scenarios, limits are unique, whereby one says that 
is the limit of ![<span style=]()
{x_n}" src="http://mathworld.wolfram.com/images/equations/Limit/Inline17.gif" style="height:14px; width:23px" /> and writes
 |
(2)
|
On the other hand, a sequence of elements from an metric 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 (1) 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 ![x=<span style=]()
{x_i}" src="http://mathworld.wolfram.com/images/equations/Limit/Inline24.gif" style="height:14px; width:45px" /> is a net from a directed set 
into 
, then an element 
is said to be the limit of 
if and only if 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 
. Similar adjustments can be made to define limits of functions 
when 
.
Limits may be taken from below
 |
(3)
|
or from above
 |
(4)
|
if the two are equal, then "the" limit is said to exist
 |
(5)
|
The expression in (2) is read "the limit as 
approaches 
from the left / from below" or "the limit as 
increases to 
," while (3) is read "the limit as 
approaches 
from the right / from above" or "the limit as 
decreases to 
." In (4), one simply refers to "the limit as 
approaches 
."
Limits are implemented in the Wolfram Language as Limit[f, x-> x0]. This command also takes options Direction (which can be set to any complex direction, including for example 
, 
, I, and -I), and Analytic, which computes symbolic limits for functions.
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 
.
A lower limit 
 |
(6)
|
is said to exist if, for every 
, 
for infinitely many values of 
and if no number less than 
has this property.
An upper limit 
 |
(7)
|
is said to exist if, for every 
, 
for infinitely many values of 
and if no number larger than 
has this property.
Related notions include supremum limit and infimum limit.
Indeterminate limit forms of types 
and 
can often be computed with L'Hospital's rule. Types 
can be converted to the form 
by writing
 |
(8)
|
Types 
, 
, and 
are treated by introducing a dependent variable
 |
(9)
|
so that
![lny=g(x)ln[f(x)],](http://mathworld.wolfram.com/images/equations/Limit/NumberedEquation10.gif) |
(10)
|
then calculating lim 
. The original limit then equals 
,
 |
(11)
|
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:
 |
(12)
|
REFERENCES:
Amend, B. Camp FoxTrot. Kansas City, MO: Andrews McMeel, p. 19, 1998.
Clark, P. L. "Convergence." 2014. http://math.uga.edu/~pete/convergence.pdf.
Courant, R. and Robbins, H. "Limits. Infinite Geometrical Series." §2.2.3 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 63-66, 1996.
Gruntz, D. On Computing Limits in a Symbolic Manipulation System. Doctoral thesis. Zürich: Swiss Federal Institute of Technology, 1996.
Hight, D. W. A Concept of Limits. New York: Prentice-Hall, 1966.
Kaplan, W. "Limits and Continuity." §2.4 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 82-86, 1992.
Miller, N. Limits: An Introductory Treatment. Waltham, MA: Blaisdell, 1964.
Mitchell, C. W. Jr. In "Media Clips" (Ed. M. Cibes and J. Greenwood). Math. Teacher 100, 339, Dec. 2006/Jan. 2007.
Munkres, J. Topology 2nd Edition. Upper Saddle River, NJ: Prentice Hall, Inc., 2000.
Nagy, G. "The Concept of Convergence: Ultrafilters and Nets." 2008. http://www.math.ksu.edu/~nagy/real-an/1-02-convergence.pdf.
Prevost, S. "Exploring the 
-
Definition of Limit with Mathematica." Mathematica Educ. 3, 17-21, 1994.
Smith, W. K. Limits and Continuity. New York: Macmillan, 1964.
Stadler, B. M. R. and Stadler, P. F. "Basic Properties of Filter Convergence Spaces." 2002. https://www.bioinf.uni-leipzig.de/~studla/Publications/PREPRINTS/01-pfs-007-subl1.pdf.
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