10.841471...0.10.998334...0.010.999983... 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 (see (ε, δ)-definition of limit below) 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 , 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., collected in Who Gave You the Epsilon?, pp. 5–13. Also available at: http://www.maa.org/pubs/Calc_articles/ma002.pdf 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.
Functions of a single variable Informally, a function has limit as approaches if approximates for near . More precisely, the value of is within a given tolerance of , provided is within a corresponding tolerance of . These two tolerances are often denoted, respectively, by (the tolerance in the value of ) and (the corresponding tolerance in ). The value of the function at is usually omitted from the approximation; for example, in many cases where limits are useful, the function has no value at (it is undefined there).
-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 (for example, the interval (0, 2)).
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 (see Cours d'Analyse). In these terms, the error (ε) in the measurement of the value at the limit can be made as small as desired by reducing the distance (δ) 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 (right) or below (left), in which case the limits may be written as
or
respectively. If these limits exist at 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 (i.e., the limit at does not exist). 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: For every , there exists a such that whenever , we have .
The limit of as approaches from below is if: For every , there exists a such that whenever , we have .
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 (e.g., by taking to be an open interval of the form ), and right-handed limits (e.g., by taking to be an open interval of the form ). It also extends the notion of one-sided limits to the included endpoints of (half-)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 (or whether) 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 (other than the existence of their non-deleted limits).
Bartle notes that although by "limit" some authors do mean this non-deleted limit, deleted limits are the most popular.For example, , , , , all take "limit" to mean the deleted limit.
Examples
Non-existence of one-sided limit(s)
The function
has no limit at (the left-hand limit does not exist due to the oscillatory nature of the sine function, and the right-hand limit does not exist due to the asymptotic behaviour of the reciprocal function, see picture), but has a limit at every other -coordinate.
The function
(a.k.a., the Dirichlet 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 equals 1 for negative and equals 2 for positive ). The limit at does not exist (the left-hand limit equals 1, whereas the right-hand limit equals 2).
Limits at only one point The functions
and
both have a limit at and it equals 0.
Limits at countably many points The function
has a limit at any -coordinate of the form where is any integer.
Limits involving infinity
Limits at infinity
Let be a function defined on The limit of as approaches infinity is , denoted
means that:
Similarly, the limit of as approaches minus infinity is , denoted
means that:
For example,
because for every , we can take such that for all real , if , then .
Another example is that
because for every , we can take such that for all real , if , then .
Infinite limits
For a function whose values grow without bound, the function diverges and the usual limit does not exist. However, in this case one may introduce limits with infinite values.
Let be a function defined on The statement the limit of as approaches is infinity, denoted
means that:
The statement the limit of as approaches is minus infinity, denoted
means that:
For example,
because for every , we can take such that for all real , if , then .
These ideas can be used together to produce definitions for different combinations, such as
or
For example,
because for every , we can take such that for all real , if , then .
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 a neighborhood of −∞ is defined to contain an interval for some a neighborhood of ∞ is defined to contain an interval where and a neighborhood of is defined in the normal way metric space In this case, is a topological space and any function of the form with 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.
Alternative notation Many authorsFor example, Limit at Encyclopedia of Mathematics allow for the projectively extended real line to be used as a way to include infinite values as well as extended real line. With this notation, the extended real line is given as and the projectively extended real line is where a neighborhood of ∞ is a set of the form The advantage is that one only needs three definitions for limits (left, right, and central) to cover all the cases. As presented above, for a completely rigorous account, we would need to consider 15 separate cases for each combination of infinities (five directions: −∞, left, central, right, and +∞; three bounds: −∞, finite, or +∞). There are also noteworthy pitfalls. For example, when working with the extended real line, does not possess a central limit (which is normal):
In contrast, when working with the projective real line, infinities (much like 0) are unsigned, so, the central limit does exist in that context:
In fact there are a plethora of conflicting formal systems in use.