Limit (mathematics)

In mathematics, a limit is the value that a function (or sequence) approaches as the argument (or index) approaches some value.^[1] 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

\lim _{x\to c}f(x)=L,

and is read as "the limit of $f$ of $x$ as $x$ approaches $c$ equals $L$ ". This means that the value of the function $f$ can be made arbitrarily close to $L$ , by choosing $x$ sufficiently close to $c$ . Alternatively, the fact that a function $f$ approaches the limit $L$ as $x$ approaches $c$ is sometimes denoted by a right arrow (→ or $\rightarrow$ ), as in

f(x)\to L{\text{ as }}x\to c,

which reads " $f$ of $x$ tends to $L$ as $x$ tends to $c$ ".

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."^[2]^[3]

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."^[4]

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.^[5]

Augustin-Louis Cauchy in 1821,^[6] 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 is due to G. H. Hardy, who introduced it in his book A Course of Pure Mathematics in 1908.^[7]

Types of limits

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.^[8]

Formally, suppose $a 1, a 2, ...$ is a sequence of real numbers. When the limit of the sequence exists, the real number $L$ is the limit of this sequence if and only if for every real number $ε > 0$ , there exists a natural number $N$ such that for all $n > N$ , we have $| a n - L | < ε$ .^[9] The common notation

\lim _{n\to \infty }a_{n}=L

is read as:

"The limit of a_n as n approaches infinity equals L" or "The limit as n approaches infinity of a_n equals L".

The formal definition intuitively means that eventually, all elements of the sequence get arbitrarily close to the limit, since the absolute value $| a n - L |$ is the distance between $a n$ and $L$ .

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 $n$ approaches infinity of a sequence ${a n}$ is simply the limit at infinity of a function $a (n)$ —defined on the natural numbers ${n}$ . On the other hand, if X is the domain of a function $f (x)$ and if the limit as $n$ approaches infinity of $f (x n)$ is $L$ for every arbitrary sequence of points ${x n}$ in X − x₀ which converges to $x 0$ , then the limit of the function $f (x)$ as $x$ approaches $x 0$ is equal to $L$ .^[10] One such sequence would be ${x 0 + 1/ n}$ .

Infinity as a limit

There is also a notion of having a limit "tend to infinity", rather than to a finite value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} . A sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{a_n\}} is said to "tend to infinity" if, for each real number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M > 0} , known as the bound, there exists an integer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} such that for each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n > N} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n > M.} That is, for every possible bound, the sequence eventually exceeds the bound. This is often written Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n\rightarrow \infty} a_n = \infty} or simply Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n \rightarrow \infty} .

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 $a_{n}=(-1)^{n}$ .

There is a corresponding notion of tending to negative infinity, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n\rightarrow \infty} a_n = -\infty} , defined by changing the inequality in the above definition to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n < M,} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M < 0.}

A sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{a_n\}} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{a_n\}_{n \geq 0}} is a sequence in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} , then the limit (when it exists) of the sequence is an element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\in M} such that, given Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon > 0} , there exists an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} such that for each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n > N} , we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(a, a_n) < \epsilon.} An equivalent statement is that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n \rightarrow a} if the sequence of real numbers $d(a,a_{n})\rightarrow 0$ .

Example: Rⁿ

An important example is the space of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -dimensional real vectors, with elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{x} = (x_1, \cdots, x_n)} where each of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i} are real, an example of a suitable distance function is the Euclidean distance, defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(\mathbf{x}, \mathbf{y}) = \|\mathbf{x} - \mathbf{y}\| = \sqrt{\sum_i(x_i - y_i)^2}.} The sequence of points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\mathbf{x}_n\}_{n \geq 0}} converges to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{x}} if the limit exists and $\|\mathbf {x} _{n}-\mathbf {x} \|\rightarrow 0$ .

Topological space

In some sense the most abstract space in which limits can be defined are topological spaces. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is a topological space with topology Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{a_n\}_{n \geq 0}} is a sequence in $X$ , then the limit (when it exists) of the sequence is a point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\in X} such that, given a (open) neighborhood Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U\in \tau} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} , there exists an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} such that for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n > N} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n \in U} is satisfied. In this case, the limit (if it exists) may not be unique. However it must be unique if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} . Given a sequence of functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{f_n\}_{n > 0}} such that each is a function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_n: E \rightarrow \mathbb{R}} , suppose that there exists a function such that for each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in E} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_n(x) \rightarrow f(x) \text{ or equivalently } \lim_{n \rightarrow \infty}f_n(x) = f(x).}

Then the sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_n} is said to converge pointwise to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} . 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f,g: E \rightarrow \mathbb{R}} is the maximum difference between the two functions as the argument Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in E} is varied. That is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(f,g) = \max_{x \in E}|f(x) - g(x)|.} Then the sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_n} is said to uniformly converge or have a uniform limit of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

A function

f (x)

for which the limit at infinity is

L

. For any arbitrary distance

ε

, there must be a value

S

such that the function stays within

L \pm ε

for all

x > S

.

Suppose $f$ is a real-valued function and $c$ is a real number. Intuitively speaking, the expression

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \to c}f(x) = L }

means that $f (x)$ can be made to be as close to $L$ as desired, by making $x$ sufficiently close to $c$ .^[11] In that case, the above equation can be read as "the limit of $f$ of $x$ , as $x$ approaches $c$ , is $L$ ".

Formally, the definition of the "limit of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} approaches Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} " is given as follows. The limit is a real number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} so that, given an arbitrary real number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon > 0} (thought of as the "error"), there is a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta > 0} such that, for any $x$ satisfying Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 < |x - c| < \delta} , it holds that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | f(x) - L | < \epsilon} . This is known as the (ε, δ)-definition of limit.

The inequality Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 < |x - c|} is used to exclude Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} from the set of points under consideration, but some authors do not include this in their definition of limits, replacing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 < |x - c| < \delta} with simply Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |x - c| < \delta} . This replacement is equivalent to additionally requiring that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} be continuous at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} .

It can be proven that there is an equivalent definition which makes manifest the connection between limits of sequences and limits of functions.^[12] The equivalent definition is given as follows. First observe that for every sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{x_n\}} in the domain of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} , there is an associated sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{f(x_n)\}} , the image of the sequence under Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} . The limit is a real number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} so that, for all sequences Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_n \rightarrow c} , the associated sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x_n) \rightarrow L} .

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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f: \mathbb{R} \rightarrow \mathbb{R}} , defined such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = 0} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \leq 0} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = 1} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x > 0} . At Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = 0} , 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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \to c^-}f(x) = 0} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \to c^+}f(x) = 1} , and from this it can be deduced Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \to c}f(x)} doesn't exist, because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \rightarrow \infty} f(x) = L.}

In this expression, the infinity is considered to be signed: either Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + \infty} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - \infty} . The "limit of f as x tends to positive infinity" is defined as follows. It is a real number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} such that, given any real Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon > 0} , there exists an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M > 0} so that if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x > M} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |f(x) - L| < \epsilon} . Equivalently, for any sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_n \rightarrow + \infty} , we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x_n) \rightarrow L} .

It is also possible to define the notion of "tending to infinity" in the value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \rightarrow c} f(x) = \infty.}

The definition is given as follows. Given any real number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M>0} , there is a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta > 0} so that for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 < |x - c| < \delta} , the absolute value of the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |f(x)| > M} . Equivalently, for any sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_n \rightarrow c} , the sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x_n) \rightarrow \infty} .

Nonstandard analysis

In non-standard analysis (which involves a hyperreal enlargement of the number system), the limit of a sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n)} can be expressed as the standard part of the value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_H} of the natural extension of the sequence at an infinite hypernatural index n=H. Thus,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=[a_n]} represented in the ultrapower construction by a Cauchy sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n)} , 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{a_n\}_{n > 0}} be a sequence in a topological space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} . For concreteness, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} can be thought of as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} , but the definitions hold more generally. The limit set is the set of points such that if there is a convergent subsequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{a_{n_k}\}_{k >0}} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{n_k}\rightarrow a} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} belongs to the limit set. In this context, such an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n = (-1)^n} . Starting from n=1, the first few terms of this sequence are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1, +1, -1, +1, \cdots} . It can be checked that it is oscillatory, so has no limit, but has limit points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{-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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma: \mathbb{R} \rightarrow X} , the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma(t)} is thought of as the "position" of the trajectory at "time" Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} . The limit set of a trajectory is defined as follows. To any sequence of increasing times Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{t_n\}} , there is an associated sequence of positions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{x_n\} = \{\gamma(t_n)\}} . If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is the limit set of the sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{x_n\}} for any sequence of increasing times, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is a limit set of the trajectory.

Technically, this is the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} -limit set. The corresponding limit set for sequences of decreasing time is called the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} -limit set.

An illustrative example is the circle trajectory: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma(t) = (\cos(t), \sin(t))} . This has no unique limit, but for each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta \in \mathbb{R}} , the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\cos(\theta), \sin(\theta))} is a limit point, given by the sequence of times Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_n = \theta + 2\pi n} . But the limit points need not be attained on the trajectory. The trajectory Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma(t) = t/(1 + t)(\cos(t), \sin(t))} also has the unit circle as its limit set.

Uses

Limits are used to define a number of important concepts in analysis.

Series

A particular expression of interest which is formalized as the limit of a sequence is sums of infinite series. These are "infinite sums" of real numbers, generally written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n = 1}^\infty a_n.} This is defined through limits as follows:^[12] given a sequence of real numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{a_n\}} , the sequence of partial sums is defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_n = \sum_{i = 1}^n a_i.} If the limit of the sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{s_n\}} exists, the value of the expression Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n = 1}^\infty a_n} is defined to be the limit. Otherwise, the series is said to be divergent.

A classic example is the Basel problem, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n = 1/n^2} . Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n = 1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}.}

However, while for sequences there is essentially a unique notion of convergence, for series there are different notions of convergence. This is due to the fact that the expression Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n = 1}^\infty a_n} does not discriminate between different orderings of the sequence $\{a_{n}\}$ , while the convergence properties of the sequence of partial sums can depend on the ordering of the sequence.

A series which converges for all orderings is called unconditionally convergent. It can be proven to be equivalent to absolute convergence. This is defined as follows. A series is absolutely convergent if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n = 1}^\infty |a_n|} is well defined. Furthermore, all possible orderings give the same value.

Otherwise, the series is conditionally convergent. A surprising result for conditionally convergent series is the Riemann series theorem: depending on the ordering, the partial sums can be made to converge to any real number, as well as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm \infty} .

Power series

A useful application of the theory of sums of series is for power series. These are sums of series of the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(z) = \sum_{n = 0}^\infty c_n z^n.} Often Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} is thought of as a complex number, and a suitable notion of convergence of complex sequences is needed. The set of values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z\in \mathbb{C}} for which the series sum converges is a circle, with its radius known as the radius of convergence.

Continuity of a function at a point

The definition of continuity at a point is given through limits.

The above definition of a limit is true even if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(c) \neq L} . Indeed, the function $f$ need not even be defined at $c$ . However, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(c)} is defined and is equal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} , then the function is said to be continuous at the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} .

Equivalently, the function is continuous at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) \rightarrow f(c)} as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \rightarrow c} , or in terms of sequences, whenever Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_n \rightarrow c} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x_n) \rightarrow f(c)} .

An example of a limit where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is not defined at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} is given below.

Consider the function

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \frac{x^2 - 1}{x - 1}.}

then $f (1)$ is not defined (see Indeterminate form), yet as $x$ moves arbitrarily close to 1, $f (x)$ correspondingly approaches 2:^[13]

$f (0.9)$	$f (0.99)$	$f (0.999)$	$f (1.0)$	$f (1.001)$	$f (1.01)$	$f (1.1)$
$1.900$	$1.990$	$1.999$	$undefined$	$2.001$	$2.010$	$2.100$

Thus, $f (x)$ can be made arbitrarily close to the limit of 2—just by making $x$ sufficiently close to $1$ .

In other words, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \to 1} \frac{x^2-1}{x-1} = 2. }

This can also be calculated algebraically, as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \frac{x^2-1}{x-1} = \frac{(x+1)(x-1)}{x-1} = x+1} for all real numbers $x \neq 1$ .

Now, since $x + 1$ is continuous in $x$ at 1, we can now plug in 1 for $x$ , leading to the equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \to 1} \frac{x^2-1}{x-1} = 1+1 = 2.}

In addition to limits at finite values, functions can also have limits at infinity. For example, consider the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \frac{2x-1}{x}} where:

$f (100) = 1.9900$
$f (1000) = 1.9990$
$f (10000) = 1.9999$

As $x$ becomes extremely large, the value of $f (x)$ approaches $2$ , and the value of $f (x)$ can be made as close to $2$ as one could wish—by making $x$ sufficiently large. So in this case, the limit of $f (x)$ as $x$ approaches infinity is $2$ , or in mathematical notation,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to\infty}\frac{2x-1}{x} = 2.}

Continuous functions

An important class of functions when considering limits are continuous functions. These are precisely those functions which preserve limits, in the sense that if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is a continuous function, then whenever Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n \rightarrow a} in the domain of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} , then the limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(a_n)} exists and furthermore is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(a)} .

In the most general setting of topological spaces, a short proof is given below:

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f: X\rightarrow Y} be a continuous function between topological spaces Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} . By definition, for each open set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} , the preimage Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(V)} is open in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} .

Now suppose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n \rightarrow a} is a sequence with limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} . Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(a_n)} is a sequence in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(a)} is some point.

Choose a neighborhood Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(a)} . Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(V)} is an open set (by continuity of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} ) which in particular contains Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} , and therefore Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(V)} is a neighborhood of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} . By the convergence of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} , there exists an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} such that for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n > N} , we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n \in f^{-1}(V)} .

Then applying Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} to both sides gives that, for the same Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} , for each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n > N} we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(a_n) \in V} . Originally Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} was an arbitrary neighborhood of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(a)} , so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(a_n) \rightarrow f(a)} . This concludes the proof.

In real analysis, for the more concrete case of real-valued functions defined on a subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E \subset \mathbb{R}} , that is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f: E \rightarrow \mathbb{R}} , a continuous function may also be defined as a function which is continuous at every point of its domain.

Limit points

In topology, limits are used to define limit points of a subset of a topological space, which in turn give a useful characterization of closed sets.

In a topological space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} , consider a subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} . A point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} is called a limit point if there is a sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{a_n\}} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S\backslash\{a\}} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n \rightarrow a} .

The reason why Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{a_n\}} is defined to be in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S\backslash\{a\}} rather than just Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is illustrated by the following example. Take Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X = \mathbb{R}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S = [0,1] \cup \{2\}} . Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \in S} , and therefore is the limit of the constant sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2, 2, \cdots} . But Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2} is not a limit point of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} .

A closed set, which is defined to be the complement of an open set, is equivalently any set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} which contains all its limit points.

Derivative

The derivative is defined formally as a limit. In the scope of real analysis, the derivative is first defined for real functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} defined on a subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E \subset \mathbb{R}} . The derivative at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in E} is defined as follows. If the limit of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{f(x+h) - f(x)}{h}} as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h \rightarrow 0} exists, then the derivative at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is this limit.

Equivalently, it is the limit as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y \rightarrow x} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{f(y) - f(x)}{y-x}.}

If the derivative exists, it is commonly denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)} .

Properties

Sequences of real numbers

For sequences of real numbers, a number of properties can be proven.^[12] Suppose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{a_n\}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{b_n\}} are two sequences converging to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} respectively.

Sum of limits is equal to limit of sum

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n + b_n \rightarrow a + b.}

Product of limits is equal to limit of product

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n \cdot b_n \rightarrow a \cdot b.}

Inverse of limit is equal to limit of inverse (as long as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \neq 0} )

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{a_n} \rightarrow \frac{1}{a}.} Equivalently, the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = 1/x} is continuous about nonzero Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} .

Cauchy sequences

A property of convergent sequences of real numbers is that they are Cauchy sequences.^[12] The definition of a Cauchy sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{a_n\}} is that for every real number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon > 0} , there is an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} such that whenever Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m, n > N} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a_m - a_n| < \epsilon.}

Informally, for any arbitrarily small error Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon} , it is possible to find an interval of diameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon} such that eventually the sequence is contained within the interval.

Cauchy sequences are closely related to convergent sequences. In fact, for sequences of real numbers they are equivalent: any Cauchy sequence is convergent.

In general metric spaces, it continues to hold that convergent sequences are also Cauchy. But the converse is not true: not every Cauchy sequence is convergent in a general metric space. A classic counterexample is the rational numbers, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Q}} , with the usual distance. The sequence of decimal approximations to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{2}} , truncated at the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} th decimal place is a Cauchy sequence, but does not converge in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Q}} .

A metric space in which every Cauchy sequence is also convergent, that is, Cauchy sequences are equivalent to convergent sequences, is known as a complete metric space.

One reason Cauchy sequences can be "easier to work with" than convergent sequences is that they are a property of the sequence $\{a_{n}\}$ alone, while convergent sequences require not just the sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{a_n\}} but also the limit of the sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} .

Order of convergence

Beyond whether or not a sequence $\{a_{n}\}$ converges to a limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} , it is possible to describe how fast a sequence converges to a limit. One way to quantify this is using the order of convergence of a sequence.

A formal definition of order of convergence can be stated as follows. Suppose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{a_n\}_{n > 0}} is a sequence of real numbers which is convergent with limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} . Furthermore, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n \neq a} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} . If positive constants Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha } exist such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty } \frac{ \left| a_{n+1} - a \right| }{ \left| a_n - a \right| ^\alpha } = \lambda } then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n } is said to converge to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a } with order of convergence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha } . The constant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda } is known as the asymptotic error constant.

Order of convergence is used for example the field of numerical analysis, in error analysis.

Computability

Limits can be difficult to compute. There exist limit expressions whose modulus of convergence is undecidable. In recursion theory, the limit lemma proves that it is possible to encode undecidable problems using limits.^[14]

There are several theorems or tests that indicate whether the limit exists. These are known as convergence tests. Examples include the ratio test and the squeeze theorem. However they may not tell how to compute the limit.

Notes

^ Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 978-0-495-01166-8.
^ Schubring, Gert (2005). Conflicts between generalization, rigor, and intuition: number concepts underlying the development of analysis in 17th-19th century France and Germany. New York: Springer. pp. 22–23. ISBN 0387228365.
^ "Euclid's Elements, Book X, Proposition 1". aleph0.clarku.edu.
^ Van Looy, Herman (1984). "A chronology and historical analysis of the mathematical manuscripts of Gregorius a Sancto Vincentio (1584–1667)". Historia Mathematica. 11 (1): 57–75. doi:10.1016/0315-0860(84)90005-3.
^ Felscher, Walter (2000), "Bolzano, Cauchy, Epsilon, Delta", American Mathematical Monthly, 107 (9): 844–862, doi:10.2307/2695743, JSTOR 2695743
^ Larson, Ron; Edwards, Bruce H. (2010). Calculus of a single variable (Ninth ed.). Brooks/Cole, Cengage Learning. ISBN 978-0-547-20998-2.
^ Miller, Jeff (1 December 2004), Earliest Uses of Symbols of Calculus, archived from the original on 2015-05-01, retrieved 2008-12-18
^ Stillwell, John (1994), Elements of algebra: geometry, numbers, equations, Springer, p. 42, ISBN 978-1441928399
^ Weisstein, Eric W. "Limit". mathworld.wolfram.com. Archived from the original on 2020-06-20. Retrieved 2020-08-18.
^ Apostol (1974, pp. 75–76)
^ Weisstein, Eric W. "Epsilon-Delta Definition". mathworld.wolfram.com. Archived from the original on 2020-06-25. Retrieved 2020-08-18.
^ ^12.0 ^12.1 ^12.2 ^12.3 Chua, Dexter. "Analysis I (based on a course given by Timothy Gowers)". Notes from the Mathematical Tripos.
^ "limit | Definition, Example, & Facts". Encyclopedia Britannica. Archived from the original on 2021-05-09. Retrieved 2020-08-18.
^ Soare, Robert I. (2014). Recursively enumerable sets and degrees : a study of computable functions and computably generated sets. Berlin: Springer-Verlag. ISBN 978-3-540-66681-3. OCLC 1154894968.

References

Apostol, Tom M. (1974), Mathematical Analysis (2nd ed.), Menlo Park: Addison-Wesley, LCCN 72011473

External links

Library resources about
Limit (mathematics)

Resources in your library

[1] Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 978-0-495-01166-8.

[2] Schubring, Gert (2005). Conflicts between generalization, rigor, and intuition: number concepts underlying the development of analysis in 17th-19th century France and Germany. New York: Springer. pp. 22–23. ISBN 0387228365.

[3] "Euclid's Elements, Book X, Proposition 1". aleph0.clarku.edu.

[4] Van Looy, Herman (1984). "A chronology and historical analysis of the mathematical manuscripts of Gregorius a Sancto Vincentio (1584–1667)". Historia Mathematica. 11 (1): 57–75. doi:10.1016/0315-0860(84)90005-3.

[5] Felscher, Walter (2000), "Bolzano, Cauchy, Epsilon, Delta", American Mathematical Monthly, 107 (9): 844–862, doi:10.2307/2695743, JSTOR 2695743

[Larson-6] Larson, Ron; Edwards, Bruce H. (2010). Calculus of a single variable (Ninth ed.). Brooks/Cole, Cengage Learning. ISBN 978-0-547-20998-2.

[7] Miller, Jeff (1 December 2004), Earliest Uses of Symbols of Calculus, archived from the original on 2015-05-01, retrieved 2008-12-18

[8] Stillwell, John (1994), Elements of algebra: geometry, numbers, equations, Springer, p. 42, ISBN 978-1441928399

[9] Weisstein, Eric W. "Limit". mathworld.wolfram.com. Archived from the original on 2020-06-20. Retrieved 2020-08-18.

[10] Apostol (1974, pp. 75–76)

[11] Weisstein, Eric W. "Epsilon-Delta Definition". mathworld.wolfram.com. Archived from the original on 2020-06-25. Retrieved 2020-08-18.

[dexter-12] 12.0 ^12.1 ^12.2 ^12.3 Chua, Dexter. "Analysis I (based on a course given by Timothy Gowers)". Notes from the Mathematical Tripos.

[13] "limit | Definition, Example, & Facts". Encyclopedia Britannica. Archived from the original on 2021-05-09. Retrieved 2020-08-18.

[14] Soare, Robert I. (2014). Recursively enumerable sets and degrees : a study of computable functions and computably generated sets. Berlin: Springer-Verlag. ISBN 978-3-540-66681-3. OCLC 1154894968.

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Limit (mathematics)

Contents

Notation

History