Limit of a function

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In mathematics, the limit of a function is a fundamental concept in analysis. Informally, a function f(x) has a limit l at a point p if the value of f(x) can be made as close to l as desired, by making x close enough to p. Formal definitions, first devised in the early 19th century, are given below.

1 History
2 Motivation
3 Definitions
4 Properties
5 See also
6 References

[edit] History

Although implicit in the development of Calculus of the 17th and 18th centuries, the modern notion of the limit of a function goes back to Bolzano who, in 1817, introduced the basics of the epsilon-delta technique (MacTutor History). However, his work was not known during his lifetime. Cauchy discussed limits in his Cours d'analyse (1821) and seems to have expressed the essence of the idea, but not in a systematic fashion (Jeff Miller). The first rigorous public presentation of the technique was given by Weierstrass in the 1850s and 1860s (MacTutor History) and has since become the standard method for dealing with limits.

The written notation using the Lim abbreviation together with the arrow below is due to Hardy in his book A Course of Pure Mathematics in 1908 (Jeff Miller).

[edit] Motivation

Imagine a traveler walking over a landscape represented by the graph of y = f(x). Her horizontal position is measured by the value of x, much like the position given by a map of the land or by a global positioning system. Her altitude is given by the coordinate y. She is walking towards the horizontal position given by x = p. As she does so, she notices that her altitude approaches l. If later asked to guess the altitude over x = p, she would then answer l, even if she had never actually reached that position.

What, then, does it mean to say that her altitude approaches l? It means that her altitude gets nearer and nearer to l except for a possible small error in accuracy. For example, suppose we set a particular accuracy goal for our traveler: she must get within a ten meters of l. She reports back that indeed she can get within ten meters of l, since she notes that when she is within fifty horizontal meters of p, her altitude is always ten meters or less from l.

We then change our accuracy goal: can she get within one meter? Yes. If she is within seven horizontal meters of p, then her altitude remains within one meter of the target l. In summary, to say that the traveler's altitude approaches l as her horizontal position approaches p means that for every target accuracy goal, there is some neighborhood of p whose altitude remains within that accuracy goal.

The initial informal statement can now be explicated:

The limit of a function f(x) as x approaches p is a number l with the following property: given any target distance from l, there is a distance from p within which the values of f(x) remain within the target distance.

This explicit statement is quite close to the formal definition of the limit of a function with values in a topological space.

[edit] Definitions

The following definitions are the generally accepted ones for the limit of a function in various contexts.

[edit] Functions on the real line

Suppose f : R → R is defined on the real line and p,l ∈ R then we say the limit of f as x approaches p is l and write

$\lim_{x \to p}f(x) = l$

if and only if for every real ε > 0 there exists a real δ > 0 such that | f(x) - l | < ε whenever 0 < | x - p | < δ. Note particularly that f(p) need not be defined.

Now x may approach p from above (right) or below (left), in which case the limits may be written as

$\lim_{x \to p^+}f(x) = l$

$\lim_{x \to p^-}f(x) = l$

respectively. If both of these limits are equal to l then this can be referred to as the limit of f(x) at p. Conversely, if they are not both equal to l then the limit, as such, does not exist.

[edit] Functions on metric spaces

Suppose f : (M,d_M) → (N,d_N) is defined between two metric spaces, with x ∈ M, p a limit point of M and l ∈ N. We say that the limit of f as x approaches p is l and write

$\lim_{x \to p}f(x) = l$

if and only if for every ε > 0 there exists a δ > 0 such that, d_N(f(x), l) < ε whenever 0 < d_M(x, p) < δ. Again, note that p need not be in the domain of f, nor does l need to be in the range of f.

An alternative definition using the concept of neighbourhood is as follows:

$\lim_{x \to p}f(x) = l$

if and only if for every neighbourhood V of l in N there exists a neighbourhood U of p in M, such that f(U - {p}) ⊆ V.

[edit] Functions on topological spaces

Suppose X,Y are topological spaces. Let p be a limit point of X, E⊆X, q ∈Y. For a function f : E → Y, we say that the limit of f as x approaches p is q (i.e., f(x)→q as x→p) and write

$\lim_{x \to p}f(x) = q$

if and only if for every neighborhood V of q, there exists a neighborhood U of p such that U∩E≠∅ and f(U∩E - {p}) ⊆ V.

[edit] Limit of a function at infinity

Limit of a function at infinity exists if, for every ε > 0 there exists an S > 0 ; so that | f(x) - L | < ε for all x > S.

If the extended real line R is considered, i.e., R ∪ {-∞, +∞}, then it is possible to define limits of a function at infinity.

Suppose f(x) is a real-valued function such that x may increase or decrease indefinitely, then we say that the limit of f as x approaches infinity is L and we write

$\lim_{x \to \infty}f(x) = L$

if and only if for every ε > 0 there exists S > 0 such that | f(x) - L | < ε whenever x > S.

Similarly, we say that the limit of f as x approaches infinity is infinity and we write

$\lim_{x \to \infty}f(x) = \infty$

if and only if for every R > 0 there exists S > 0 such that for all real numbers f(x) > R whenever x > S.

In an analogous way, the following expressions can be defined:

$\lim_{x \to -\infty}f(x) = L, \lim_{x \to \infty}f(x) = -\infty, \lim_{x \to -\infty}f(x) = \infty, \lim_{x \to -\infty}f(x) = -\infty$ .

These notions of a limit attempt to provide a metric space interpretation to limits at infinity. However, note that these notions of a limit are consistent with the topological space definition of limit if

a neighborhood of -∞ is defined to contain the interval [-∞,c) where c∈R
a neighborhood of ∞ is defined to contain the interval (c,∞] where c∈R
a neighborhood of a∈R is defined in the normal way metric space R

In this case, R is a topological space and any function of the form f:X → Y with X,Y⊆ R is subject to the topological definition of a limit. Note that with this topological definition, it is easy to define infinite limits at finite points, which have not been defined above in the metric sense.

[edit] Evaluating limits at infinity for rational functions

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.

[edit] Complex-valued functions

The complex plane with metric $d (x, y): = | x - y |$ is also a metric space. There are two different types of limits when we consider complex-valued functions.

[edit] Limit of a function at a point

Suppose f is a complex-valued function, then we write

$\lim_{x \to p}f(x) = L$

if and only if

for every ε > 0 there exists a δ >0 such that for all real numbers x with 0<|x-p|<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.

[edit] Limit of a function of more than one variable

By noting that |x-p| represents a distance, the definition of a limit can be extended to functions of more than one variable. In the case of a function f : R² → R,

$\lim_{(x,y) \to (p, q)} f(x, y) = L$

if and only if

for every ε > 0 there exists a δ > 0 such that for all (x,y) with 0 < ||(x,y)-(p,q)|| < δ, we have |f(x,y)-L| < ε

where ||(x,y)-(p,q)|| represents the Euclidean distance. This can be extended to any number of variables.

[edit] 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 with limit equal to p, the sequence (f(x_n)) converges with limit L.

If the sets A, B, ... form a finite partition of the function domain, $x\in\overline{A} \land x\in\overline{B}$ , ... and the relative limit for each of those sets exist and is the equal to, say, L, then the limit exists for the point x and is equal to L.

The function f is continuous at p if and only if the limit of f(x) as x approaches p exists and is finite. 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, provided the limits on the right sides of the identity below exist:

$\begin{matrix} \lim\limits_{x \to p} & (f(x) + g(x)) & = & \lim\limits_{x \to p} f(x) + \lim\limits_{x \to p} g(x) \\ \lim\limits_{x \to p} & (f(x) - g(x)) & = & \lim\limits_{x \to p} f(x) - \lim\limits_{x \to p} g(x) \\ \lim\limits_{x \to p} & (f(x) g(x)) & = & \lim\limits_{x \to p} f(x) \cdot \lim\limits_{x \to p} g(x) \\ \lim\limits_{x \to p} & (f(x)/g(x)) & = & {\lim\limits_{x \to p} f(x) / \lim\limits_{x \to p} g(x)} \end{matrix}$

(the last provided that the denominator is non-zero). In each case above, when the limits on the right do not exist, or, in the last case, when the limits in both the numerator and the denominator are zero, nonetheless the limit on the left may still exist -- this depends on which functions f and g are.

These rules are also valid for one-sided 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. Indeterminate forms — for instance, 0/0, 0×∞, ∞−∞, and ∞/∞ — are also not covered by these rules, but the corresponding limits can often be determined with L'Hôpital's rule.

[edit] See also

[edit] References

Visual Calculus by Lawrence S. Husch, University of Tennessee (2001)

Apostol, Tom M., Mathematical Analysis, 2nd ed. Addison-Wesley, 1974. ISBN 0201002884.

Sutherland, W. A., Introduction to Metric and Topological Spaces. Oxford University Press, Oxford, 1975. ISBN 0 19 853161 3.

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Category: Calculus

Limit of a function

From Wikipedia, the free encyclopedia

Contents

[edit] History

[edit] Motivation

[edit] Definitions

[edit] Functions on the real line

[edit] Functions on metric spaces

[edit] Functions on topological spaces

[edit] Limit of a function at infinity

[edit] Evaluating limits at infinity for rational functions

[edit] Complex-valued functions

[edit] Limit of a function at a point

[edit] Limit of a function of more than one variable

[edit] Properties

[edit] See also

[edit] References

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