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GeezerVison is showing mathworld.wolfram.com — Limit -- from Wolfram MathWorld 
LimitA limit describes the value approached by a sequence or function.
A sequence
where
Limits in a metric space are unique. On the other hand, a sequence in a topological
space
The topological notion of convergence can be rewritten to accommodate a wider array of topological spaces
For a function The limit laws describe how finite limits behave under algebraic operations.
For a real-valued function
For a domain
or from above
If
The expression in (3) is read "the limit as
For a real function
where
For a function of two real variables, a limit must be independent of the path of approach. For
example, the function
for
In contrast, the function
Consequently, its limit at the origin does not exist. The relation
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
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
For a sequence
These values always exist in the affinely extended real numbers and may equal Related notions include supremum limit and infimum limit.
Indeterminate forms of types
For
so that
then calculating
The indeterminate form
All of the above notions can be generalized even further by utilizing the language of ultrafilters. In particular, if 
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:
See alsoCentral Limit Theorem, Continuous, Convergent, Deleted Neighborhood, Derivative, Discontinuity, Indeterminate Form, Infimum Limit, L'Hospital's Rule, Limit Comparison Test, Limit Laws, Limit Test, Lower Limit, Squeeze Theorem, Supremum Limit, Upper Limit Explore this topic in the MathWorld classroomPortions of this entry contributed by Christopher Stover Explore with Wolfram|Alpha
More things to try: ReferencesAmend, B. Camp FoxTrot. Kansas City, MO: Andrews McMeel, p. 19, 1998. Strip reproduced with permission of the author.Clark, P. L. "Convergence." 2014. https://web.archive.org/web/20170706035307/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, Switzerland: Swiss Federal Institute of Technology, 1996.Hight, D. W. A Concept of Limits. New York: Prentice-Hall, 1966.Hunter, J. K. An Introduction to Real Analysis. Davis, CA: University of California, 2014. https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/intro_analysis.pdf.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, 2006/2007.Munkres, J. Topology 2nd Edition. Upper Saddle River, NJ: Prentice Hall, Inc., 2000.Nagy, G. "The Concept of Convergence: Ultrafilters and Nets." 2008. https://www.math.ksu.edu/about/personal/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.Referenced on Wolfram|AlphaLimitCite this as:Weisstein, Eric W., with contributions by Christopher Stover. "Limit." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Limit.html Subject classifications | ||||||||||||||||||||||||||||||||||||||
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of 
is said to have limit 
provided that for each 
of 
,
so that 
.
is a 
in 
has limit 
if 
,
so that
is the 
.
is the limit of 
and writes
may have several, even infinitely many,
different limits provided that 
is equipped with a 
approaches 
is 
.
by utilizing the language of 
is a 
into 
,
is said to be the limit of 
of 
,
is eventually in 
,
so that, for every 
with 
,
lies in 
.
with 
,
be an 
.
if, 
,
for every 
with 
.
and would not specify a 
with 
and an 
of 
,
approaches 
is 
(respectively 
)
(respectively 
),
(respectively 
)
with 
(Hunter 2014, p. 118).
,
to be an 
(Hunter 2014, pp. 116-117). Limits may be taken from
below
is an 
and 
,
approaches 
from the left / from below" or "the limit as 
increases to 
,
approaches 
from the right / from above" or "the limit as 
decreases to 
.
approaches 
.
defined for all sufficiently large 
,
is its limit at 
and 
range over the 
is chosen so that 
is defined for every 
.
,
by 
and require 
to be defined for every 
.
satisfies the 
.
tends to 0 as 
approaches the 
has different limits along the 
and 
:
also shows that agreement along both coordinate 
,
,
in a 
is nothing more than a 
mapping 
to 
.
of 
and 
are defined in the 
or 
.
is the least 
has 
for infinitely many 
,
is the greatest such 
and 
can often be computed with 
can be converted to the form 
by writing
,
,
,
are treated by introducing a 
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.
,
is also frequently encountered.
is a 
is an 
,
is said to be a limit of 
if every 
belongs to 
.