Limit (mathematics)
In mathematics, a limit is the value that a function "approaches" as the input "approaches" some value. Limits are essential to calculus and are used to define continuity, derivatives, and integrals.
The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory.
In formulas, a limit of a function is usually written as
and is read as "the limit of of as approaches equals ". The fact that a function approaches the limit as approaches is sometimes denoted by a right arrow, as in
Limit of a function
Suppose is a real-valued function and is a real number. Intuitively speaking, the expressionmeans that can be made to be as close to as desired by making sufficiently close to. In that case, the above equation can be read as "the limit of of, as approaches, is ".
Augustin-Louis Cauchy in 1821, followed by Karl Weierstrass, formalized the definition of the limit of a function which became known as the -definition of limit. The definition uses to represent any small positive number, so that " becomes arbitrarily close to " means that eventually lies in the interval, which can also be written using the absolute value sign as. The phrase "as approaches " then indicates that we refer to values of whose distance from is less than some positive number —that is, values of within either or, which can be expressed with. The first inequality means that the distance between and is greater than and that, while the second indicates that is within distance of.
The above definition of a limit is true even if. Indeed, the function need not even be defined at.
For example, if
then is not defined, yet as moves arbitrarily close to 1, correspondingly approaches 2:
f | f | f | f | f | f | f |
1.900 | 1.990 | 1.999 | undefined | 2.001 | 2.010 | 2.100 |
Thus, can be made arbitrarily close to the limit of 2 just by making sufficiently close to.
In other words,
This can also be calculated algebraically, as for all real numbers.
Now since is continuous in at 1, we can now plug in 1 for, thus.
In addition to limits at finite values, functions can also have limits at infinity. For example, consider
- f = 1.9900
- f = 1.9990
- f = 1.9999
Limit of a sequence
Consider the following sequence: 1.79, 1.799, 1.7999,... It can be observed that the numbers are "approaching" 1.8, the limit of the sequence.Formally, suppose is a sequence of real numbers. It can be stated that the real number is the limit of this sequence, namely:
which is read as
to mean
Intuitively, this means that eventually all elements of the sequence get arbitrarily close to the limit, since the absolute value is the distance between and. Not every sequence has a limit; if it does, it is called convergent, and if it does not, it is 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 the one hand, the limit as approaches infinity of a sequence is simply the limit at infinity of a function defined on the natural numbers. On the other hand, if is the domain of a function and if the limit as approaches infinity of is for every arbitrary sequence of points in which converges to, then the limit of the function as approaches is. One such sequence would be.
Limit as "standard part"
In non-standard analysis, the limit of a sequence can be expressed as the standard part of the value of the natural extension of the sequence at an infinite hypernatural index n=H. Thus,Here the standard part function "st" rounds off each finite hyperreal number to the nearest real number. This formalizes the natural intuition that for "very large" values of the index, the terms in the sequence are "very close" to the limit value of the sequence. Conversely, the standard part of a hyperreal represented in the ultrapower construction by a Cauchy sequence, is simply the limit of that sequence:
In this sense, taking the limit and taking the standard part are equivalent procedures.
Convergence and fixed point
A formal definition of convergence can be stated as follows.Suppose as goes from to is a sequence that converges to, with for all. If positive constants and exist with
then as goes from to converges to of order, with asymptotic error constant
Given a function with a fixed point, there is a nice checklist for checking the convergence of the sequence.
then there is linear convergence | |
series diverges | |
then there is at least linear convergence and maybe something better, the expression should be checked for quadratic convergence |
then there is quadratic convergence provided that is continuous | |
then there is something even better than quadratic convergence | |
does not exist | then there is convergence that is better than linear but still not quadratic |