Limit (mathematics)
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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 larger and larger; or the behavior of a sequence's elements, as their index becomes larger and larger. Limits are used in calculus and other branches of mathematical analysis to define derivatives and continuity.
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.
Mathematics students usually first encounter limits in introductory calculus classes, and understanding the detailed concept often presents a stumbling block. Readers seeking an introductory explanation might look at the Wikibooks Calculus section about limits [1]. This present article does have some elementary exposition but it's also about how limits are treated in more advanced branches of mathematics.
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Limit of a function
- Main article: limit of a function
Suppose f(x) is a real function and c is a real number. The expression:
- <math> \lim_{x \to c}f(x) = L </math>
means that f(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 f(x), as x approaches c, is L". Note that this statement can be true even if f(c) <math>\neq</math> L. Indeed, the function f(x) need not even be defined at c. Two examples help illustrate this.
Consider <math> f(x) = \frac{x}{x^2 + 1} </math> 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 | <math>\Rightarrow</math> 0.4 <math>\Leftarrow</math> | 0.3998 | 0.3988 | 0.3882 |
As x approaches 2, f(x) approaches 0.4 and hence we have <math>\lim_{x\to 2}f(x)=0.4</math>. In the case where <math>f(c) = \lim_{x\to c} f(x)</math>, f is said to be continuous at x = c. But it is not always the case. Consider
- <math>g(x)=\left\{\begin{matrix} \frac{x}{x^2+1}, & \mbox{if }x\ne 2 \\ \\ 0, & \mbox{if }x=2. \end{matrix}\right.</math>
The limit of g(x) as x approaches 2 is 0.4 (just as in f(x)), but <math>\lim_{x\to 2}g(x)\neq g(2)</math>; g is not continuous at x = 2.
Or, consider the case where f(x) is undefined at x = c.
- <math> f(x) = \frac{x - 1}{\sqrt{x} - 1} </math>
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 | <math>\Rightarrow</math> undef <math>\Leftarrow</math> | 2.001 | 2.010 | 2.10 |
Thus, x can get as close to 1, so long as it is not equal to 1, so that the limit of <math> f(x) </math> is 2.
Formal definition
A limit is formally defined as follows: Let <math>f</math> be a function defined on an open interval containing <math>c</math> (except possibly at <math>c</math>) and let <math>L</math> be a real number. The statement
- <math> \lim_{x \to c}f(x) = L </math>
means that for each <math> \varepsilon\ >0</math> there exists a <math> \delta\ >0</math> such that for all <math> x </math> where <math>0<|x-c|< \delta\ </math>, then <math>| f (x)-L|< \varepsilon\ </math>.
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 number; rather, it means that x either becomes larger and larger (for positive infinity) or smaller and smaller (for negative infinity).
For example, consider <math>f(x) = \frac{2x}{x + 1}</math>.
- f(100) = 1.9802
- f(1000) = 1.9980
- f(10000) = 1.9998
As x becomes extremely large, f(x) approaches 2. In this case,
- <math> \lim_{x \to \infty} f(x) = 2 </math>
If one considers the codomain of f is the extension real line, then limit of a function at infinity could be considered as a special case of limit of a function at a point.
- <math> \lim_{x \to \infty} f(x) = c </math> if and only if for each <math> \epsilon > 0 \quad\exists n </math> such that <math>|f(x) - c| < \epsilon </math> whenever <math> x > n </math>
Limit of a sequence
Main article: 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.
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
- <math> \lim_{n \to \infty} x_n = L </math>
if and only if
- for every 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| can be interpreted as 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.
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).
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.
An alternative is the concept of limit for filters on topological spaces.
Limit in category theory
Main article: limit (category theory)
See also
ca:Límit cs:Limita de:Limes (Mathematik) es:Límite matemático eo:Limeso fr:Limite (mathématiques) ko:극한 io:Limito id:Limit it:Limite (matematica) he:גבול (מתמטיקה) lmo:Límit (matemàtica) hu:Határérték nl:Limiet ja:極限 no:Grenseverdi pl:Granica (matematyka) pt:Limite fi:Raja-arvo sv:Gränsvärde tr:Limit uk:Границя zh:极限