Wikibooks Calculus has a page on the topic of In mathematics, the concept of a "limit" is used to describe the behavior of a function as its argument either "gets close" to some point, or as it becomes arbitrarily large; or the behavior of a sequence's elements as their index increases indefinitely. Limits are used in calculus and other branches of mathematical analysis to define derivatives and continuity. Image File history File links Wikibooks-logo-en. ...
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Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...
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For other senses of this word, see sequence (disambiguation). ...
In mathematics, an index is a superscript or subscript to a symbol. ...
For other uses, see Calculus (disambiguation). ...
Analysis has its beginnings in the rigorous formulation of calculus. ...
For a non-technical overview of the subject, see Calculus. ...
In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...
The concept of the "limit of a function" is further generalized to the concept of topological net, while the limit of a sequence is closely related to limit and direct limit in category theory. In mathematics the term net has at least two meanings. ...
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts of mathematics, like products and inverse limits. ...
In mathematics, the direct limit (also called the inductive limit) is a general method of taking limits of directed families of objects. We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category. ...
In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...
Limit of a function
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Suppose ƒ(x) is a real-valued function and c is a real number. The expression: In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ...
In mathematics, a function of a real variable is a mathematical function whose domain is the real line. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
 means that ƒ(x) can be made to be as close to L as desired by making x sufficiently close to c. In that case, we say that "the limit of ƒ of x, as x approaches c, is L". Note that this statement can be true even if  . Indeed, the function ƒ(x) need not even be defined at c. Two examples help illustrate this.
Consider f(x) = x/(x2 + 1) as x approaches 2. In this case, f(x) is defined at 2 and equals its limit of 0.4:
| f(1.9) | f(1.99) | f(1.999) | f(2) | f(2.001) | f(2.01) | f(2.1) | | 0.4121 | 0.4012 | 0.4001 |  0.4  | 0.3998 | 0.3988 | 0.3882 | As x approaches 2, ƒ(x) approaches 0.4 and hence we have  . In the case where  , ƒ is said to be continuous at x = c. But it is not always the case. Consider In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...
 The limit of g(x) as x approaches 2 is 0.4 (just as in ƒ(x)), but  ; g is not continuous at x = 2.
Or, consider the case where ƒ(x) is undefined at x = c.
 In this case, as x approaches 1, f(x) is undefined at x = 1 but the limit equals 2: | f(0.9) | f(0.99) | f(0.999) | f(1.0) | f(1.001) | f(1.01) | f(1.1) | | 1.95 | 1.99 | 1.999 | undef | 2.001 | 2.010 | 2.10 | Thus, f(x) can be made arbitrarily close to the limit of 2 just by making x sufficiently close enough to 1.
Formal definition Karl Weierstrass formally defined a limit as follows: Karl Theodor Wilhelm Weierstrass (Weierstraß) (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). ...
Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
means that - for each real ε > 0 there exists a real δ > 0 such that for all x with 0 < |x − c| < δ, we have |f(x) − L| < ε.
The formal definition of a limit is sometimes called the epsilon-delta form because it uses the Greek letters delta (δ) and epsilon (ε). The use of the particular Greek letters δ and ε is merely traditional; the definition would, of course, be unchanged if different letters or symbols were used. The Greek alphabet (Greek: ) is an alphabet consisting of 24 letters that has been used to write the Greek language since the late 8th or early 8th century BC. It was the first alphabet in the narrow sense, that is, a writing system using a separate symbol for each vowel...
Look up Δ, δ in Wiktionary, the free dictionary. ...
Look up Ε, ε in Wiktionary, the free dictionary. ...
Caution: It should be noted that this definition provides a way to recognize a limit without providing a way to calculate it. One often needs to find a limit using informal methods, especially when f(x) is discontinuous at c, for example, when f is a ratio with a denominator that becomes 0 at c. One should check that the result actually meets the Weierstrass definition in such cases.
Limit of a function at infinity A related concept to limits as x approaches some finite number is the limit as x approaches positive or negative infinity. This does not literally mean that the difference between x and infinity becomes small, since infinity is not a real number; rather, it means that x either grows without bound positively (positive infinity) or grows without bound negatively (negative infinity). For other uses, see Infinity (disambiguation). ...
For example, consider f(x) = 2x/(x + 1).
- f(100) = 1.9802
- f(1000) = 1.9980
- f(10000) = 1.9998
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 just by picking x sufficiently large. In this case, we say that the limit of f(x) as x approaches infinity is 2. In mathematical notation,  Formally, we have the definition  if and only if for each ε > 0 there exists an n such that  Note that the n in the definition will generally depend on ε. A similar definition applies for 
If one considers the domain of f to be the extended real number line, then the limit of a function at infinity can be considered as a special case of limit of a function at a point. In mathematics, the domain of a function is the set of all input values to the function. ...
The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ...
Limit of a sequence -
Consider the following sequence: 1.79, 1.799, 1.7999,... We could observe that the numbers are "approaching" 1.8, the limit of the sequence. The limit of a sequence is one of the oldest concepts in mathematical analysis. ...
Formally, suppose x1, x2, ... is a sequence of real numbers. We say that the real number L is the limit of this sequence and we write For other senses of this word, see sequence (disambiguation). ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
 if and only if ↔ ⇔ ≡ logical symbols representing iff. ...
- for every real number ε > 0 there exists a natural number n0 (which will depend on ε) such that for all n > n0 we have |xn − L| < ε.
Intuitively, this means that eventually all elements of the sequence get as close as we want to the limit, since the absolute value |xn − L| is the distance between xn and L. Not every sequence has a limit; if it does, we call it convergent, otherwise divergent. One can show that a convergent sequence has only one limit. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...
In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ...
The limit of a sequence and the limit of a function are closely related. On one hand, the limit of a sequence is simply the limit at infinity of a function defined on natural numbers. On the other hand, a limit of a function f at x, if it exists, is the same as the limit of the sequence xn = f(x + 1/n). In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...
Useful Identities In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). ...
In calculus, lHôpitals rule uses derivatives to help compute limits with indeterminate forms. ...
Topological net -
Main article: Net (topology) All of the above notions of limit can be unified and generalized to arbitrary topological spaces by introducing topological nets and defining their limits. The article on nets elaborates on this. In mathematics the term net has at least two meanings. ...
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In mathematics the term net has at least two meanings. ...
An alternative is the concept of limit for filters on topological spaces. In mathematics, a filter is a special subset of a partially ordered set. ...
Limit in category theory -
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts of mathematics, like products and inverse limits. ...
See also In mathematics, a one-sided limit is where the limit of a function is defined in moving in the positive or negative direction, but not both. ...
In calculus, lHôpitals rule uses derivatives to help compute limits with indeterminate forms. ...
In calculus, the squeeze theorem (also known as the pinching theorem or the sandwich theorem, sometimes the squeeze lemma) is a theorem regarding the limit of a function. ...
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, where S is a 
, if the denominator containing the limit does not equal zero
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