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
for all . In many commonly-encountered scenarios, limits are unique, whereby one says that is the limit of and writes
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 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
or from above
if the two are equal, then "the" limit is said to exist
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
is said to exist if, for every , for infinitely many values of and if no number less than has this property.
An upper limit
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 infimumlimit.
Indeterminate limit forms of types and can often be computed with L'Hospital's rule. Types can be converted to the form by writing
Types , , and are treated by introducing a dependent variable
then calculating lim . The original limit then equals ,
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: