Limit of a function




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Although the function is not defined at zero, as becomes closer and closer to zero, becomes arbitrarily close to 1. In other words, the limit of as approaches zero, equals 1.



In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input which may or may not be in the domain of the function.
Formal definitions, first devised in the early 19th century, are given below. Informally, a function assigns an output to every input. We say that the function has a limit at an input, if gets closer and closer to as moves closer and closer to. More specifically, the output value can be made arbitrarily close to if the input to is taken sufficiently close to. On the other hand, if some inputs very close to 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.

History

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 Bernard 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. Bruce Pourciau argues that Isaac Newton, in his 1687 Principia, demonstrates a more sophisticated understanding of limits than he is generally given credit for, including being the first to present an epsilon argument.
In his 1821 book Cours d'analyse, Augustin-Louis Cauchy discussed variable quantities, infinitesimals and limits, and defined continuity of by saying that an infinitesimal change in necessarily produces an infinitesimal change in, while Grabiner claims that he used a rigorous epsilon-delta definition in proofs. In 1861, Karl Weierstrass first introduced the epsilon-delta definition of limit in the form it is usually written today. He also introduced the notations and
The modern notation of placing the arrow below the limit symbol is due to G. H. Hardy, which is introduced in his book A Course of Pure Mathematics in 1908.

Motivation

Imagine a person walking on a landscape represented by the graph. Their horizontal position is given by, much like the position given by a map of the land or by a global positioning system. Their altitude is given by the coordinate. Suppose they walk towards a position, as they get closer and closer to this point, they will notice that their altitude approaches a specific value. If asked about the altitude corresponding to, they would reply by saying.
What, then, does it mean to say, their altitude is approaching ? It means that their altitude gets nearer and nearer to —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. They report back that indeed, they can get within ten vertical meters of, arguing that as long as they are within fifty horizontal meters of, their altitude is always within ten meters of.
The accuracy goal is then changed: can they get within one vertical meter? Yes, supposing that they are able to move within five horizontal meters of, their altitude will always remain within one meter from the target altitude. Summarizing the aforementioned concept we can say that the traveler's altitude approaches as their horizontal position approaches, so as to say that for every target accuracy goal, however small it may be, there is some neighbourhood of, within which for every member the target accuracy goal is fulfilled by an altidude , except for maybe the horizontal position itself.
The initial informal statement can now be explicated:
In fact, this explicit statement is quite close to the formal definition of the limit of a function, with values in a topological space.
More specifically, to say that
is to say that can be made as close to as desired, by making close enough, but not equal, to .
The following definitions, known as -definitions, are the generally accepted definitions for the limit of a function in various contexts.

Functions of a single variable

-definition of limit

Suppose is a function defined on the real line, and there are two real numbers and. One would say: "The limit of of, as approaches, exists, and it equals ". and write,
or alternatively, say " tends to as tends to ", and write,
if the following property holds: for every real, there exists a real such that for all real, implies. Symbolically,
For example, one may say
because for every real, we can take, so that for all real, if, then.
A more general definition applies for functions defined on subsets of the real line. Let be a subset of Let be a real-valued function. Let be a point such that there exists some open interval containing with It is then said that the limit of as approaches is, if:
Symbolically,
For example, one may say
because for every real, we can take, so that for all real, if, then. In this example, contains open intervals around the point 1.
Here, note that the value of the limit does not depend on being defined at, nor on the value —if it is defined. For example, let
because for every, we can take, so that for all real, if, then. Note that here is undefined.
In fact, a limit can exist in which equals where is the interior of, and are the isolated points of the complement of. In our previous example where We see, specifically, this definition of limit allows a limit to exist at 1, but not at 0 or 2.
The letters and can be understood as "error" and "distance". In fact, Cauchy used as an abbreviation for "error" in some of his work, though in his definition of continuity, he used an infinitesimal rather than either or . In these terms, the error in the measurement of the value at the limit can be made as small as desired by reducing the distance to the limit point. As discussed below, this definition also works for functions in a more general context. The idea that and represent distances helps suggest these generalizations.

Existence and one-sided limits

Alternatively, may approach from above or below, in which case the limits may be written as
or
respectively. If these limits exist at p and are equal there, then this can be referred to as the limit of at. If the one-sided limits exist at, but are unequal, then there is no limit at . If either one-sided limit does not exist at, then the limit at also does not exist.
A formal definition is as follows. The limit of as approaches from above is if:
The limit of as approaches from below is if:
If the limit does not exist, then the oscillation of at is non-zero.

More general definition using limit points and subsets

Limits can also be defined by approaching from subsets of the domain.
In general: Let be a real-valued function defined on some Let be a limit point of some —that is, is the limit of some sequence of elements of distinct from. Then we say the limit of, as approaches from values in, is , written
if the following holds:
Note, can be any subset of, the domain of. And the limit might depend on the selection of. This generalization includes as special cases limits on an interval, as well as left-handed limits of real-valued functions, and right-handed limits. It also extends the notion of one-sided limits to the included endpoints of closed intervals, so the square root function can have limit 0 as approaches 0 from above:
since for every, we may take such that for all, if, then.
This definition allows a limit to be defined at limit points of the domain, if a suitable subset which has the same limit point is chosen.
Notably, the previous two-sided definition works on which is a subset of the limit points of.
For example, let The previous two-sided definition would work at but it wouldn't work at 0 or 2, which are limit points of.

Deleted versus non-deleted limits

The definition of limit given here does not depend on how is defined at. Bartle refers to this as a deleted limit, because it excludes the value of at. The corresponding non-deleted limit does depend on the value of at, if is in the domain of. Let be a real-valued function. The non-deleted limit of, as approaches, is if
The definition is the same, except that the neighborhood now includes the point, in contrast to the deleted neighborhood. 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.
Bartle notes that although by "limit" some authors do mean this non-deleted limit, deleted limits are the most popular.

Examples

Non-existence of one-sided limit(s)

The function
has no limit at , but has a limit at every other -coordinate.
The function
has no limit at any -coordinate.

Non-equality of one-sided limits

The function
has a limit at every non-zero -coordinate. The limit at does not exist.

Limits at only one point

The functions
and
both have a limit at and it equals 0.