Limit of a function
In
mathematics, the
limit of a function is a fundamental concept in
mathematical analysis.
Rather informally, to say that a function f has a limit y when x tends to a value x_{0} (or to the infinity), is to say that the values taken by the expression f(x) get close to y when x gets close to x_{0} (or gets infinitely big). Formal definitions, first devised around the end of the 19th century, are given below.
See net (topology) for a generalisation of the concept of limit.
See mathematical analysis.
Suppose f : (M,d_{M}) > (N,d_{N}) is a map between two metric spaces, p∈M and L∈N. We say that "the limit of f(x) is L as x approaches p" and write
if and only if
 for every ε > 0 there exists a δ > 0 such that for all x in M with 0 < d_{M}(x, p) < δ, we have d_{N}(f(x), L) < ε.
Realvalued functions
The real number line is itself a metric space. But it has some different types of limits.
Suppose f is a realvalued function, then we write

if and only if
 for every ε > 0 there exists a δ >0 such that for all real numbers x with 0<xp<δ, we have f(x)L<ε
It is just a particular case of functions on metric spaces, with both M and N are the real numbers.
Or we write

if and only if
 for every R > 0 there exists a δ >0 such that for all real numbers x with 0<xp<δ, we have f(x)>R;
or we write

if and only if
 for every R < 0 there exists a δ >0 such that for all real numbers x with 0<xp<δ, we have f(x)<R;
If, in the definitions, xp is used instead of xp, then we get a righthanded limit, denoted by lim_{x→p+}. If px is used, we get a lefthanded limit, denoted by lim_{x→p}.
Suppose f(x) is a realvalued function. We can also consider the limit of function when x increases or decreases indefinitely.
We write

if and only if
 for every ε > 0 there exists S >0 such that for all real numbers x>S, we have f(x)L<ε
or we write

 for every R > 0 there exists S >0 such that for all real numbers x>S, we have f(x)>R;.
Similarly, we can define .
There are three basic rules for evaluating limits at infinity for a rational function f(x) = p(x)/q(x):
 If the degree of p is greater than the degree of q, then the limit is positive or negative infinity depending on the signs of the leading coefficients
 If the degree of p and q are equal, the limit is the leading coefficient of p divided by the leading coefficient of q
 If the degree of p is less than the degree of q, the limit is 0
If the limit at infinity exists, it represents a horizontal asymptote at x = L. Polynomials do not have horizontal asymptotes; they may occur with rational functions.
The complex plane is also a metric space. There are two different types of limits when we consider complexvalued functions.
Suppose f is a complexvalued function, then we write

if and only if
 for every ε > 0 there exists a δ >0 such that for all real numbers x with 0<xp<δ, we have f(x)L<ε
It is just a particular case of functions over metric spaces with both M and N are the complex plane.
We write

if and only if
 for every ε > 0 there exists S >0 such that for all complex numbers x>S, we have f(x)L<ε
Examples
Functions on metric spaces
 If z is a complex number with z < 1, then the sequence z, z^{2}, z^{3}, ... of complex numbers converges with limit 0. Geometrically, these numbers "spiral into" the origin, following a logarithmic spiral.
 In the metric space C[a,b] of all continuous functions defined on the interval [a,b], with distance arising from the supremum norm, every element can be written as the limit of a sequence of polynomial functions. This is the content of the StoneWeierstrass theorem.
Properties
To say that the limit of a function f at p is L is equivalent to saying
 for every convergent sequence (x_{n}) in M  {p} with limit equal to p, the sequence (f(x_{n})) converges with limit L.
In the case that
f is realvalued, then it is also equivalent to saying that both the righthanded limit or lefthanded limit of
f at
p are
L.
The function f is continuous at p if and only if the limit of f(x) as x approaches p exists and is equal to f(p). Equivalently, f transforms every sequence in M which converges towards p into a sequence in N which converges towards f(p).
Again, if N is a normed vector space, then the limit operation is linear in the following sense: if the limit of f(x) as x approaches p is L and the limit of g(x) as x approaches p is P, then the limit of f(x) + g(x) as x approaches p is L + P. If a is a scalar from the base field, then the limit of af(x) as x approaches p is aL.
Taking the limit of functions is compatible with the algebraic operations:
If

and
then
and
and
(the latter provided that
f_{2}(x) is nonzero in a
neighborhood of
p and
L_{2} is nonzero as well).
These rules are also valid for onesided limits, for the case p = ±∞, and also for infinite limits using the rules
 q + ∞ = ∞ for q ≠ ∞
 q × ∞ = ∞ if q > 0
 q × ∞ = ∞ if q < 0
 q / ∞ = 0 if q ≠ ± ∞
(see
extended real number line).
Note that there is no general rule for the case q / 0; it all depends on the way 0 is approached. Interdeterminate forms, for instance 0/0, 0×∞ ∞∞ or ∞/∞, are also not covered by these rules but the corresponding limits can usually be determined with l'Hôpital's rule.
References