Limit (mathematics)
In
mathematics, the concept of a "
limit" is used to describe the
behavior of a
function, as its
argument gets "close" to either some
point, or
infinity; or the behavior of a
sequence's elements, as their
index approaches infinity. 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.
Main article: limit of a function
Suppose f(x) is a real function and c is a real number. If the values of f(x) approach (get close to, but don't necessarily reach) the number L, as x approaches c, one can state that "the limit of f(x), as x approaches c, is L" and write
For example: f(x)=x/(x^{2}+1). f(1.9)=0.4121, f(1.99)=0.4012; f(1.999)=0.4001. As x approaches 2, f(x) approaches 0.4 and hence we have lim_{x→2} f(x)=0.4.
In this case lim_{x→2} f(x)=f(2) and f is continuous at x=2. But it is not always the case,
consider

Then lim
_{x→2} g(
x)≠
g(2) and so
g is not continuous at
x=2.
One need not examine limits only as x approches some finite number; one can also examine the limit, of a function, as x approaches infinity. For example
f(
x)=2
x /
x+1.
f(100)=1.9802,
f(1000)=1.9980,
f(10000)=1.9998. As
x becomes extremely large,
f(
x) approaches 2. In this case:
However, 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.
Main article: limit of a sequence
Consider the following sequence: 1.79, 1.799, 1.7999,... We could observe that
the numbers are "approaching" the 1.8, the limit of the sequence.
Formally, suppose x_{1}, x_{2}, ... is a sequence of real numbers.
We say that the real number L is the limit of this sequence and we write
if and only if
 for every ε>0 there exists a natural number n_{0} (which will depend on ε) such that for all n>n_{0} we have x_{n}  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 
x_{n} 
L can be interpreted as the "distance" between
x_{n} 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 land, a limit of a function f at x, if it exists, is the same as the limit of the sequence x_{n}=f(x+1/n).
Main article: net (topology)
Better introduction is needed
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.
Main article: limit (category theory)
A introduction will be added soon.