In mathematics, a limit is the value that a function (or sequence) approaches as the argument (or index) approaches some value. Limits of functions are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals. The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory. The limit inferior and limit superior provide generalizations of the concept of a limit which are particularly relevant when the limit at a point may not exist.
Notation In formulas, a limit of a function is usually written as
and is read as "the limit of of as approaches equals ". This means that the value of the function can be made arbitrarily close to , by choosing sufficiently close to . Alternatively, the fact that a function approaches the limit as approaches is sometimes denoted by a right arrow (→ or ), as in
or in
which reads " of tends to as tends to ".
History
According to Hankel (1871), the modern concept of limit originates from Proposition X.1 of Euclid's Elements, which forms the basis of the Method of exhaustion found in Euclid and Archimedes: "Two unequal magnitudes being set out, if from the greater there is subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process is repeated continually, then there will be left some magnitude less than the lesser magnitude set out."
Grégoire de Saint-Vincent gave the first definition of limit (terminus) of a geometric series in his work Opus Geometricum (1647): "The terminus of a progression is the end of the series, which none progression can reach, even not if she is continued in infinity, but which she can approach nearer than a given segment."
In the Scholium to in 1687, Isaac Newton had a clear definition of a limit, stating that "Those ultimate ratios ... are not actually ratios of ultimate quantities, but limits ... which they can approach so closely that their difference is less than any given quantity". Bruce Pourciau further argues that, in addition to Newton actually having a more sophisticated understanding of limits than he is generally credited with, he also provided the first epsilon argument.
The modern definition of a limit goes back to Bernard Bolzano who, in 1817, developed the basics of the epsilon-delta technique to define continuous functions. However, his work remained unknown to other mathematicians until thirty years after his death.
Augustin-Louis Cauchy in 1821, followed by Karl Weierstrass, formalized the definition of the limit of a function which became known as the (ε, δ)-definition of limit.
The modern notation of placing the arrow below the limit symbol was invented by John Gaston Leathem in 1905 and popularized by G. H. Hardy's 1908 textbook A Course of Pure Mathematics.
Types of limits
In sequences
Real numbers A sequence of real numbers is said to converge to a limit if the terms of the sequence give an arbitrarily good approximation of the number , after discarding finitely many initial terms. An example is the sequence and so on. Each term in the sequence is an approximation to the rational number . If an approximation is desired that lies within a given error , then all of the terms of the sequence except finitely many lie within the target error. For example, to get within of , the approximations , , and are outside the error window, but after discarding those three approximations, of the remaining approximations are within the error range . The same approximation property holds for every positive error .
More precisely, a sequence of real numbers is said to be convergent if there exists an such that for each positive real number , there exists a positive integer (depending on ) such that every term with is within distance of . The number is unique, when it exists. It is called the limit of the sequence , and this is written
The definition of the limit of a sequence can thus be summarized:
If such an does not exist, is said to be divergent.
A sequence is said to tend to infinity if all but finitely many terms exceed each given number. An example is the sequence of positive integers. This tends to infinity because, for any given bound , every element of the sequence is greater than after discarding the finite number that are not. Formally, tends to infinity if, given any real , there exists a positive integer such that for every . This is written as .
Similarly, the sequence is said to tend to negative infinity if all but finitely many terms are less than any given lower bound. This is written as .
In each case where or , the sequence is said to have an infinite limit. But it does not have a limit in the sense of being convergent to a real number, since and are not real numbers.
Metric space The discussion of sequences above is for sequences of real numbers. The notion of limits can be defined for sequences valued in more abstract spaces, such as metric spaces. If is a metric space with distance function and is a sequence in then the limit (when it exists) of the sequence is an element such that, given there exists an such that for each we have
An equivalent statement is that if the sequence of real numbers
Example: ℝn An important example is the space of -dimensional real vectors, with elements where each of the are real, an example of a suitable distance function is the Euclidean distance, defined by
The sequence of points converges to if the limit exists and
Topological space In some sense the most abstract space in which limits can be defined are topological spaces. If is a topological space with topology and is a sequence in then the limit (when it exists) of the sequence is a point such that, given a (open) neighborhood of there exists an such that for every
is satisfied. In this case, the limit (if it exists) may not be unique. However it must be unique if is a Hausdorff space.
Function space This section deals with the idea of limits of sequences of functions, not to be confused with the idea of limits of functions, discussed below.
The field of functional analysis partly seeks to identify useful notions of convergence on function spaces. For example, consider the space of functions from a generic set to Given a sequence of functions such that each is a function suppose that there exists a function such that for each
Then the sequence is said to converge pointwise to However, such sequences can exhibit unexpected behavior. For example, it is possible to construct a sequence of continuous functions which has a discontinuous pointwise limit.
Another notion of convergence is uniform convergence. The uniform distance between two functions is the maximum difference between the two functions as the argument is varied. That is,
Then the sequence is said to uniformly converge or have a uniform limit of if with respect to this distance. The uniform limit has "nicer" properties than the pointwise limit. For example, the uniform limit of a sequence of continuous functions is continuous.
Many different notions of convergence can be defined on function spaces. This is sometimes dependent on the regularity of the space. Prominent examples of function spaces with some notion of convergence are Lp spaces and Sobolev space.
In functions
Suppose is a real-valued function and is a real number. Intuitively speaking, the expression
means that can be made to be as close to as desired, by making sufficiently close to . In that case, the above equation can be read as "the limit of of , as approaches , is ".
Formally, the definition of the "limit of as approaches " is given as follows. The limit is a real number so that, given an arbitrary real number (thought of as the "error"), there is a such that, for any satisfying it holds that This is known as the -definition of limit.
The inequality is used to exclude from the set of points under consideration, but some authors do not include this in their definition of limits, replacing with simply This replacement is equivalent to additionally requiring that be continuous at
It can be proven that there is an equivalent definition which makes manifest the connection between limits of sequences and limits of functions. The equivalent definition is given as follows. First observe that for every sequence in the domain of , there is an associated sequence the image of the sequence under The limit is a real number so that, for all sequences the associated sequence
One-sided limit
It is possible to define the notion of having a "left-handed" limit ("from below"), and a notion of a "right-handed" limit ("from above"). These need not agree. An example is given by the positive indicator function, defined such that if and if At the function has a "left-handed limit" of 0, a "right-handed limit" of 1, and its limit does not exist. Symbolically, this can be stated as, for this example, and and from this it can be deduced does not exist, because
Infinity in limits of functions It is possible to define the notion of "tending to infinity" in the domain of
This could be considered equivalent to the limit as a reciprocal tends to 0:
or it can be defined directly: the "limit of as tends to positive infinity" is defined as a value such that, given any real there exists an so that for all The definition for sequences is equivalent: As we have
In these expressions, the infinity is normally considered to be signed ( or and corresponds to a one-sided limit of the reciprocal. A two-sided infinite limit can be defined, but an author would explicitly write to be clear.
It is also possible to define the notion of "tending to infinity" in the value of
Again, this could be defined in terms of a reciprocal:
Or a direct definition can be given as follows: given any real number there is a so that for the absolute value of the function A sequence can also have an infinite limit: as the sequence
This direct definition is easier to extend to one-sided infinite limits. While mathematicians do talk about functions approaching limits "from above" or "from below", there is not a standard mathematical notation for this as there is for one-sided limits.
Nonstandard analysis In non-standard analysis (which involves a hyperreal enlargement of the number system), the limit of a sequence can be expressed as the standard part of the value of the natural extension of the sequence at an infinite hypernatural index . Thus,
Here, the standard part function "st" rounds off each finite hyperreal number to the nearest real number (the difference between them is infinitesimal). This formalizes the natural intuition that for "very large" values of the index, the terms in the sequence are "very close" to the limit value of the sequence. Conversely, the standard part of a hyperreal represented in the ultrapower construction by a Cauchy sequence is simply the limit of that sequence:
In this sense, taking the limit and taking the standard part are equivalent procedures.
Limit sets Limit set of a sequence Let be a sequence in a topological space For concreteness, can be thought of as but the definitions hold more generally. The limit set is the set of points such that if there is a convergent subsequence with then belongs to the limit set. In this context, such an is sometimes called a limit point.
A use of this notion is to characterize the "long-term behavior" of oscillatory sequences. For example, consider the sequence Starting from n=1, the first few terms of this sequence are It can be checked that it is oscillatory, so has no limit, but has limit points
Limit set of a trajectory