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Limit


The term limit comes about relative to a number of topics from several different branches of mathematics.

A sequence x_1,x_2,... of elements in a topological space X is said to have limit x provided that for each neighborhood U of x, there exists a natural number N so that x_n in U for all n>=N. This very general definition can be specialized in the event that X is a metric space, whence one says that a sequence {x_n} in X has limit L if for all epsilon>0, there exists a natural number n_0 in N so that

 |x_n-L|<epsilon
(1)

for all n>=n_0. In many commonly-encountered scenarios, limits are unique, whereby one says that L is the limit of {x_n} and writes

 L=lim_(n->infty)x_n.
(2)

On the other hand, a sequence of elements from an metric space X may have several - even infinitely many - different limits provided that X is equipped with a topology which fails to be T2. One reads the expression in (1) as "the limit as n approaches infinity of x_n is L."

The topological notion of convergence can be rewritten to accommodate a wider array of topological spaces X by utilizing the language of nets. In particular, if x={x_i} is a net from a directed set I into X, then an element x in X is said to be the limit of x if and only if for every neighborhood U of x, x is eventually in U, i.e., if there exists an i in I so that, for every j in I with j>=i, the point x_j lies in U. This notion is particularly well-purposed for topological spaces which aren't first-countable.

A function f(z) is said to have a finite limit c=lim_(z->a)f(z) if, for all epsilon>0, there exists a delta>0 such that |f(z)-c|<epsilon whenever 0<|z-a|<delta. 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 f(z) as z approaches a is equal to +infty (respectively -infty) if for every number N>0 (respectively N<0), there exists a number delta depending on N for which f(z)>N (respectively, f(z)<N) whenever 0<|z-a|<delta. Similar adjustments can be made to define limits of functions f(z) when z->+/-infty.

Limits may be taken from below

 lim_(z->a^-)=lim_(z^a)
(3)

or from above

 lim_(z->a^+)=lim_(zva).
(4)

if the two are equal, then "the" limit is said to exist

 lim_(z->a)=lim_(z->a^-)=lim_(z->a^+).
(5)

The expression in (2) is read "the limit as z approaches a from the left / from below" or "the limit as z increases to a," while (3) is read "the limit as z approaches a from the right / from above" or "the limit as z decreases to a." In (4), one simply refers to "the limit as z approaches a."

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 +1, -1, 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 x_1,x_2,... in a topological space X is nothing more than a function g:N->X mapping n to x_n.

A lower limit h

 lowerlim_(n->infty)S_n=lim_(n->infty)__S_n=h
(6)

is said to exist if, for every epsilon>0, |S_n-h|<epsilon for infinitely many values of n and if no number less than h has this property.

An upper limit k

 upperlim_(n->infty)S_n=lim_(n->infty)^_S_n=k
(7)

is said to exist if, for every epsilon>0, |S_n-k|<epsilon for infinitely many values of n and if no number larger than k has this property.

Related notions include supremum limit and infimum limit.

Indeterminate limit forms of types infty/infty and 0/0 can often be computed with L'Hospital's rule. Types 0·infty can be converted to the form 0/0 by writing

 f(x)g(x)=(f(x))/(1/g(x)).
(8)

Types 0^0, infty^0, and 1^infty are treated by introducing a dependent variable

 y=f(x)^(g(x))
(9)

so that

 lny=g(x)ln[f(x)],
(10)

then calculating lim lny. The original limit then equals e^(limlny),

 L=limf(x)^(g(x))=e^(limlny).
(11)

The indeterminate form infty-infty is also frequently encountered.

All of the above notions can be generalized even further by utilizing the language of ultrafilters. In particular, if (X,T) is a topological space and if U is an ultrafilter on X, then an element x in X is said to be a limit of U if every neighborhood of x belongs to U. 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:

 lim_(x->infty)(sqrt(x^3-x^2+3x))/(sqrt(x^3)-sqrt(x^2)+sqrt(3x))=1.
(12)
FoxTrot by Bill Amend, June 2, 1996 strip. Reproduced with permission of the author.

See also

Central Limit Theorem, Continuous, Convergent, Derivative, Discontinuity, Indeterminate, Infimum Limit, L'Hospital's Rule, Limit Comparison Test, Limit Test, Lower Limit, Squeeze Theorem, Supremum Limit, Upper Limit Explore this topic in the MathWorld classroom

This entry contributed by Christopher Stover

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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 epsilon-delta 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|Alpha

Limit

Cite this as:

Stover, Christopher. "Limit." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Limit.html

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