Rather informally, to say that a function f has a limit y when x tends to a value x0 (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 x0 (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.
Table of contents |
2 Formal definition 3 Examples 4 Properties 5 See also 6 References |
Suppose f : (M,dM) -> (N,dN) 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
The real number line is itself a metric space. But it has some different types of limits.
Suppose f is a real-valued function, then we write
Or we write
Suppose f(x) is a real-valued function. We can also consider the limit of function when x increases or decreases indefinitely.
We write
There are three basic rules for evaluating limits at infinity for a rational function f(x) = p(x)/q(x):
The complex plane is also a metric space. There are two different types of limits when we consider complex-valued functions.
Suppose f is a complex-valued function, then we write
We write
To say that the limit of a function f at p is L is equivalent to saying
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
These rules are also valid for one-sided limits, for the case p = ±∞, and also for infinite limits using the rules
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.
History
Formal definition
Functions on metric spaces
if and only ifReal-valued functions
Limit of a function at a point
if and only if
It is just a particular case of functions on metric spaces, with both M and N are the real numbers.
if and only if
or we write
if and only if
If, in the definitions, x-p is used instead of |x-p|, then we get a right-handed limit, denoted by limx→p+. If p-x is used, we get a left-handed limit, denoted by limx→p-.Limit of function at infinity
if and only if
or we write
Similarly, we can define .
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.Complex-valued functions
Limit of a function at a point
if and only if
It is just a particular case of functions over metric spaces with both M and N are the complex plane.Limit of a function at infinity
if and only if
Examples
Real-valued functions
Functions on metric spaces
Properties
In the case that f is real-valued, then it is also equivalent to saying that both the right-handed limit or left-handed limit of f at p are L.
and
then
and
and
(the latter provided that f2(x) is non-zero in a neighborhood of p and L2 is non-zero as well).
(see extended real number line).See also
References