In sequences Real numbers The expression
0.999... should be interpreted as the limit of the sequence 0.9, 0.99, 0.999, ... and so on. This sequence can be rigorously shown to have the limit 1, and therefore this expression is meaningfully interpreted as having the value 1. Formally, suppose is a
sequence of
real numbers. When the limit of the sequence exists, the real number is the
limit of this sequence
if and only if for every
real number , there exists a
natural number such that for all , we have . The common notation \lim_{n \to \infty} a_n = L is read as: or The formal definition intuitively means that eventually, all elements of the sequence get arbitrarily close to the limit, since the
absolute value is the distance between and . Not every sequence has a limit. A sequence with a limit is called
convergent; otherwise it is called
divergent. One can show that a convergent sequence has only one limit. The limit of a sequence and the limit of a function are closely related. On one hand, the limit as approaches infinity of a sequence is simply the limit at infinity of a function —defined on the
natural numbers . On the other hand, if is the domain of a function and if the limit as approaches infinity of is for
every arbitrary sequence of points in which converges to , then the limit of the function as approaches is equal to . One such sequence would be .
Infinity as a limit There is also a notion of having a limit "tend to infinity", rather than to a finite value A sequence \{a_n\} is said to "tend to infinity" if, for each real number known as the bound, there exists an integer N such that for each a_n > M. That is, for every possible bound, the sequence eventually exceeds the bound. This is often written \lim_{n\rightarrow \infty} a_n = \infty or simply It is possible for a sequence to be divergent, but not tend to infinity. Such sequences are called
oscillatory. An example of an oscillatory sequence is There is a corresponding notion of tending to negative infinity, {{nowrap|\lim_{n\rightarrow \infty} a_n = -\infty,}} defined by changing the inequality in the above definition to a_n with M A sequence \{a_n\} with \lim_{n\rightarrow \infty} |a_n| = \infty is called
unbounded, a definition equally valid for sequences in the
complex numbers, or in any
metric space. Sequences which do not tend to infinity are called
bounded. Sequences which do not tend to positive infinity are called
bounded above, while those which do not tend to negative infinity are
bounded below.
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 M is a metric space with distance function and \{a_n\}_{n \geq 0} is a sequence in then the limit (when it exists) of the sequence is an element a\in M such that, given there exists an N such that for each we have d(a, a_n) An equivalent statement is that a_n \rightarrow a if the sequence of real numbers
Example: n An important example is the space of n-dimensional real vectors, with elements \mathbf{x} = (x_1, \cdots, x_n) where each of the x_i are real, an example of a suitable distance function is the
Euclidean distance, defined by d(\mathbf{x}, \mathbf{y}) = \|\mathbf{x} - \mathbf{y}\| = \sqrt{\sum_i(x_i - y_i)^2}. The sequence of points \{\mathbf{x}_n\}_{n \geq 0} converges to \mathbf{x} if the limit exists and {{nowrap|\|\mathbf{x}_n - \mathbf{x}\| \rightarrow 0.}}
Topological space In some sense the
most abstract space in which limits can be defined are
topological spaces. If X is a topological space with topology and \{a_n\}_{n \geq 0} is a sequence in then the limit (when it exists) of the sequence is a point a\in X such that, given a (open)
neighborhood U\in \tau of there exists an N such that for every a_n \in U is satisfied. In this case, the limit (if it exists) may not be unique. However it must be unique if X 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 E to {{nowrap|\mathbb{R}.}} Given a sequence of functions \{f_n\}_{n > 0} such that each is a function {{nowrap|f_n: E \rightarrow \mathbb{R},}} suppose that there exists a function such that for each f_n(x) \rightarrow f(x) \text{ or equivalently } \lim_{n \rightarrow \infty}f_n(x) = f(x). Then the sequence f_n 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 f,g: E \rightarrow \mathbb{R} is the maximum difference between the two functions as the argument x \in E is varied. That is, d(f,g) = \max_{x \in E}|f(x) - g(x)|. Then the sequence f_n is said to
uniformly converge or have a
uniform limit of f if f_n \rightarrow f 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 is . For any arbitrary distance , there must be a value such that the function stays within for all .|upright=1.4 Suppose is a
real-valued function and is a
real number. Intuitively speaking, the expression \lim_{x \to c}f(x) = L 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 f(x) as x approaches c" is given as follows. The limit is a real number L so that, given an arbitrary real number \varepsilon > 0 (thought of as the "error"), there is a \delta > 0 such that, for any x satisfying it holds that This is known as the
-definition of limit. The inequality 0 is used to exclude c from the set of points under consideration, but some authors do not include this in their definition of limits, replacing 0 with simply This replacement is equivalent to additionally requiring that f 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 \{x_n\} in the domain of f, there is an associated sequence {{nowrap|\{f(x_n)\},}} the image of the sequence under The limit is a real number L 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, {{nowrap|f: \mathbb{R} \rightarrow \mathbb{R},}} defined such that f(x) = 0 if and f(x) = 1 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, {{nowrap|\lim_{x \to c^-}f(x) = 0,}} and {{nowrap|\lim_{x \to c^+}f(x) = 1,}} and from this it can be deduced \lim_{x \to c}f(x) does not exist, because {{nowrap|\lim_{x \to c^-}f(x) \neq \lim_{x \to c^+}f(x).}}
Infinity in limits of functions It is possible to define the notion of "tending to infinity" in the domain of \lim_{x \rightarrow +\infty} f(x) = L. This could be considered equivalent to the limit as a reciprocal tends to 0: \lim_{x' \rightarrow 0^+} f(1/x') = L. or it can be defined directly: the "limit of f as x tends to positive infinity" is defined as a value L such that, given any real there exists an M > 0 so that for all The definition for sequences is equivalent: As we have In these expressions, the infinity is normally considered to be signed (+\infty 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 \pm\infty to be clear. It is also possible to define the notion of "tending to infinity" in the value of \lim_{x \rightarrow c} f(x) = \infty. Again, this could be defined in terms of a reciprocal: \lim_{x \rightarrow c} \frac{1}{f(x)} = 0. Or a direct definition can be given as follows: given any real number there is a \delta > 0 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 (a_n) can be expressed as the
standard part of the value a_H of the natural extension of the sequence at an infinite
hypernatural index . Thus, \lim_{n \to \infty} a_n = \operatorname{st}(a_H) . 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 a=[a_n] represented in the ultrapower construction by a Cauchy sequence is simply the limit of that sequence: \operatorname{st}(a)=\lim_{n \to \infty} a_n . In this sense, taking the limit and taking the standard part are equivalent procedures.
Limit sets Limit set of a sequence Let \{a_n\}_{n > 0} be a sequence in a topological space For concreteness, X can be thought of as {{nowrap|\mathbb{R},}} but the definitions hold more generally. The
limit set is the set of points such that if there is a convergent
subsequence \{a_{n_k}\}_{k >0} with {{nowrap|a_{n_k}\rightarrow a,}} then a belongs to the limit set. In this context, such an a 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 {{nowrap|\{-1, +1\}.}}
Limit set of a trajectory This notion is used in
dynamical systems, to study limits of trajectories. Defining a trajectory to be a function {{nowrap|\gamma: \mathbb{R} \rightarrow X,}} the point \gamma(t) is thought of as the "position" of the trajectory at "time" The limit set of a trajectory is defined as follows. To any sequence of increasing times {{nowrap|\{t_n\},}} there is an associated sequence of positions {{nowrap|\{x_n\} = \{\gamma(t_n)\}.}} If x is the limit set of the sequence \{x_n\} for any sequence of increasing times, then x is a limit set of the trajectory. Technically, this is the \omega-limit set. The corresponding limit set for sequences of decreasing time is called the \alpha-limit set. An illustrative example is the circle trajectory: This has no unique limit, but for each {{nowrap|\theta \in \mathbb{R},}} the point (\cos(\theta), \sin(\theta)) is a limit point, given by the sequence of times But the limit points need not be attained on the trajectory. The trajectory \gamma(t) = t/(1 + t)(\cos(t), \sin(t)) also has the
unit circle as its limit set. == Uses ==