Limit of a function
Although the function (sin x)/x is not defined at zero, as x becomes closer and closer to zero, (sin x)/x becomes arbitrarily close to 1. In other words, the limit of (sin x)/x, as x approaches zero, equals 1.
Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f(x) to every input x. We say that the function has a limit L at an input p, if f(x) gets closer and closer to L as x moves closer and closer to p. More specifically, when f is applied to any input sufficiently close to p, the output value is forced arbitrarily close to L. On the other hand, if some inputs very close to p are taken to outputs that stay a fixed distance apart, then we say the limit does not exist.
The notion of a limit has many applications in modern calculus. In particular, the many definitions of continuity employ the concept of limit: roughly, a function is continuous if all of its limits agree with the values of the function. The concept of limit also appears in the definition of the derivative: in the calculus of one variable, this is the limiting value of the slope of secant lines to the graph of a function.
Although implicit in the development of calculus of the 17th and 18th centuries, the modern idea of the limit of a function goes back to Bolzano who, in 1817, introduced the basics of the epsilon-delta technique to define continuous functions. However, his work was not known during his lifetime.
Imagine a person walking over a landscape represented by the graph of y = f(x). Their 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. Their altitude is given by the coordinate y. They walk toward the horizontal position given by x = p. As they get closer and closer to it, they notice that their altitude approaches L. If asked about the altitude of x = p, they would then answer L.
What, then, does it mean to say, their altitude is approaching L? It means that their 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: they must get within ten meters of L. They report back that indeed, they can get within ten vertical meters of L, since they note that when they are within fifty horizontal meters of p, their altitude is always ten meters or less from L.
The accuracy goal is then changed: can they get within one vertical meter? Yes. If they are anywhere within seven horizontal meters of p, their altitude will always remain within one meter from the target L. In summary, to say that the traveler's altitude approaches L as their horizontal position approaches p, is to say that for every target accuracy goal, however small it may be, there is some neighbourhood of p whose altitude fulfills that accuracy goal.
In fact, this explicit statement is quite close to the formal definition of the limit of a function, with values in a topological space.
is to say that ƒ(x) can be made as close to L as desired, by making x close enough, but not equal, to p.
The following definitions, known as (ε, δ)-definitions, are the generally accepted definitions for the limit of a function in various contexts.
Suppose f : R → R is defined on the real line and p, L ∈ R. One would say that the limit of f, as x approaches p, is L and written
A more general definition applies for functions defined on subsets of the real line. Let (a, b) be an open interval in R, and p a point of (a, b). Let f be a real-valued function defined on all of (a, b)—except possibly at p itself. It is then said that the limit of f as x approaches p is L, if for every real ε > 0, there exists a real δ > 0 such that 0 < | x − p | < δ and x ∈ (a, b) implies that | f(x) − L | < ε.
Here, note that the value of the limit does not depend on f being defined at p, nor on the value f(p)—if it is defined.
Alternatively, x may approach p from above (right) or below (left), in which case the limits may be written as
respectively. If these limits exist at p and are equal there, then this can be referred to as the limit of f(x) at p. If the one-sided limits exist at p, but are unequal, then there is no limit at p (i.e., the limit at p does not exist). If either one-sided limit does not exist at p, then the limit at p also does not exist.
A formal definition is as follows. The limit of f(x) as x approaches p from above is L if, for every ε > 0, there exists a δ > 0 such that |f(x) − L| < ε whenever 0 < x − p < δ. The limit of f(x) as x approaches p from below is L if, for every ε > 0, there exists a δ > 0 such that |f(x) − L| < ε whenever 0 < p − x < δ.
If the limit does not exist, then the oscillation of f at p is non-zero.
Apart from open intervals, limits can be defined for functions on arbitrary subsets of R, as follows (Bartle & Sherbert 2000): let f be a real-valued function defined on a subset S of the real line. Let p be a limit point of S—that is, p is the limit of some sequence of elements of S distinct from p. The limit of f, as x approaches p from values in S, is L, if for every ε > 0, there exists a δ > 0 such that 0 < |x − p| < δ and x ∈ S implies that |f(x) − L| < ε.
The definition of limit given here does not depend on how (or whether) f is defined at p. Bartle (1967) refers to this as a deleted limit, because it excludes the value of f at p. The corresponding non-deleted limit does depend on the value of f at p, if p is in the domain of f:
The definition is the same, except that the neighborhood | x − p | < δ now includes the point p, in contrast to the deleted neighborhood 0 < | x − p | < δ. This makes the definition of a non-deleted limit less general. One of the advantages of working with non-deleted limits is that they allow to state the theorem about limits of compositions without any constraints on the functions (other than the existence of their non-deleted limits) (Hubbard (2015)).
Bartle (1967) notes that although by "limit" some authors do mean this non-deleted limit, deleted limits are the most popular. For example, Apostol (1974), Courant (1924), Hardy (1921), Rudin (1964), Whittaker & Watson (1902) all take "limit" to mean the deleted limit.
has a limit at every non-zero x-coordinate (the limit equals 1 for negative x and equals 2 for positive x). The limit at x = 0 does not exist (the left-hand limit equals 1, whereas the right-hand limit equals 2).
Suppose M and N are subsets of metric spaces A and B, respectively, and f : M → N is defined between M and N, with x ∈ M, p a limit point of M and L ∈ N. It is said that the limit of f as x approaches p is L and write
Again, note that p need not be in the domain of f, nor does L need to be in the range of f, and even if f(p) is defined it need not be equal to L.
An alternative definition using the concept of neighbourhood is as follows:
Suppose X,Y are topological spaces with Y a Hausdorff space. Let p be a limit point of Ω ⊆ X, and L ∈Y. For a function f : Ω → Y, it is said that the limit of f as x approaches p is L (i.e., f(x) → L as x → p) and written
This last part of the definition can also be phrased "there exists an open punctured neighbourhood U of p such that f(U∩Ω) ⊆ V ".
Alternatively, the requirement that Y be a Hausdorff space can be relaxed to the assumption that Y be a general topological space, but then the limit of a function may not be unique. In particular, one can no longer talk about the limit of a function at a point, but rather a limit or the set of limits at a point.
A function is continuous at a limit point p of and in its domain if and only if f(p) is the (or, in the general case, a) limit of f(x) as x tends to p.
Similarly, the limit of f as x approaches negative infinity is L, denoted
These ideas can be combined in a natural way to produce definitions for different combinations, such as
Limits involving infinity are connected with the concept of asymptotes.
These notions of a limit attempt to provide a metric space interpretation to limits at infinity. In fact, they are consistent with the topological space definition of limit if
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.
In contrast, when working with the projective real line, infinities (much like 0) are unsigned, so, the central limit does exist in that context:
There are three basic rules for evaluating limits at infinity for a rational function f(x) = p(x)/q(x): (where p and q are polynomials):
If the limit at infinity exists, it represents a horizontal asymptote at y = L. Polynomials do not have horizontal asymptotes; such asymptotes may however occur with rational functions.
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,
where ||(x,y) − (p,q)|| represents the Euclidean distance. This can be extended to any number of variables.
Let f : X → Y be a mapping from a topological space X into a Hausdorff space Y, p ∈ X a limit point of X and L ∈ Y.
If L is the limit (in the sense above) of f as x approaches p, then it is a sequential limit as well, however the converse need not hold in general. If in addition X is metrizable, then L is the sequential limit of f as x approaches p if and only if it is the limit (in the sense above) of f as x approaches p.
For functions on the real line, one way to define the limit of a function is in terms of the limit of sequences. (This definition is usually attributed to Eduard Heine.) In this setting:
The notion of the limit of a function is very closely related to the concept of continuity. A function ƒ is said to be continuous at c if it is both defined at c and its value at c equals the limit of f as x approaches c:
If a function f is real-valued, then the limit of f at p is L if and only if both the right-handed limit and left-handed limit of f at p exist and are 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 equal to f(p). If f : M → N is a function between metric spaces M and N, then it is equivalent that f transforms every sequence in M which converges towards p into a sequence in N which converges towards f(p).
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.
These rules are also valid for one-sided limits, including when p is ∞ or −∞. In each rule above, when one of the limits on the right is ∞ or −∞, the limit on the left may sometimes still be determined by the following rules.
In other cases the limit on the left may still exist, although the right-hand side, called an indeterminate form, does not allow one to determine the result. This depends on the functions f and g. These indeterminate forms are:
As an example of this phenomenon, consider the following functions that violates both additional restrictions:
Thus, the naïve chain rule would suggest that the limit of f(f(x)) is 0. However, it is the case that
This rule uses derivatives to find limits of indeterminate forms 0/0 or ±∞/∞, and only applies to such cases. Other indeterminate forms may be manipulated into this form. Given two functions f(x) and g(x), defined over an open interval I containing the desired limit point c, then if:
Specifying an infinite bound on a summation or integral is a common shorthand for specifying a limit.