epsilon-delta

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.

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.

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.

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 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.

In his 1821 book '' Cours d'analyse'', Cauchy discussed variable quantities, infinitesimals and limits, and defined continuity of $y=f(x)$ by saying that an infinitesimal change in ''x'' necessarily produces an infinitesimal change in ''y'', while claims that he used a rigorous epsilon-delta definition in proofs., collected i
Who Gave You the Epsilon?
pp. 5–13. Also available at: http://www.maa.org/pubs/Calc_articles/ma002.pdf In 1861, Weierstrass first introduced the epsilon-delta definition of limit in the form it is usually written today. He also introduced the notations lim and lim_{''x''→''x''0}.

The modern notation of placing the arrow below the limit symbol is due to Hardy, which is introduced in his book '' A Course of Pure Mathematics'' in 1908.

Motivation

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.

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.

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

:$\lim_f(x) = L$,

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.

Functions of a single variable

(''ε'', ''δ'')-definition of limit

Suppose $f: \R \rightarrow \R$ is a function defined on the real line, and there are two real numbers ''p'' and ''L''. One would say that the limit of ''f'', as ''x'' approaches ''p'', is ''L'' and written

:$\lim_ f(x) = L$,

or alternatively, say ''f''(''x'') tends to ''L'' as ''x'' tends to ''p'', and written:

:$f(x) \to L \;\; \text \;\; x \to p$,

if the following property holds:

For every real , there exists a real such that for all real ''x'', implies .

Or, symbolically:
:$(\forall \varepsilon > 0 ) \, (\exists \delta > 0) \, (\forall x \in \R) \, (0 < , x - p, < \delta \implies , f(x) - L, < \varepsilon)$.

For example, we may say
:$\lim_ 4x + 1 = 9$
because for every real ''ε'' > 0, we can take ''δ'' = ''ε''/4, so that for all real ''x'', if 0 < , ''x'' − ''p'', < ''δ'', then , ''f''(''x'') − ''L'', < ''ε''.

A more general definition applies for functions defined on subsets of the real line. Let (''a'', ''b'') be an open interval in $\R$, and a number ''p'' in (''a'', ''b''). Let $f: S \to \R$ be a real-valued function defined on ''S'' — a set contains 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 , there exists a real such that for all , implies that .

Or, symbolically:
:$(\forall \varepsilon > 0 ) \, (\exists \delta > 0) \, (\forall x \in (a, b)) \, (0 < , x - p, < \delta \implies , f(x) - L, < \varepsilon)$.

For example, we may say
:$\lim_ \sqrt = 2$
because for every real ''ε'' > 0, we can take ''δ'' = ''ε'', so that for all real ''x'' ≥ −3, if 0 < , ''x'' − 1, < ''δ'', then , ''f''(''x'') − 2, < ''ε''. In this example, ''S'' = [−3, ∞) 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 ''f'' being defined at ''p'', nor on the value ''f''(''p'')—if it is defined. For example,
:$\lim_ \frac = 3$
because for every ''ε'' > 0, we can take ''δ'' = ''ε''/2, so that for all real ''x'' ≠ 1, if 0 < , ''x'' − 1, < ''δ'', then , ''f''(''x'') − 3, < ''ε''. Note that here ''f''(1) is undefined.

The letters ''ε'' and ''δ'' can be understood as "error" and "distance". In fact, Cauchy used ''ε'' as an abbreviation for "error" in some of his work, though in his definition of continuity, he used an infinitesimal $\alpha$ rather than either ''ε'' or ''δ'' (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 ''δ'' and ''ε'' represent distances helps suggest these generalizations.

Existence and one-sided limits

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

:$\lim_f(x) = L$

or

:$\lim_f(x) = L$

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'' as ''x'' approaches ''p'' from above is ''L'' if:
:For every ''ε'' > 0, there exists a ''δ'' > 0 such that whenever 0 < ''x'' − ''p'' < ''δ'', we have , ''f''(''x'') − ''L'',  < ''ε''.
:$(\forall \varepsilon > 0 ) \, (\exists \delta > 0) \, (\forall x \in (a,b))\, (0 < x - p < \delta \implies , f(x) - L, < \varepsilon)$.

The limit of ''f'' as ''x'' approaches ''p'' from below is ''L'' if:
:For every ''ε'' > 0, there exists a ''δ'' > 0 such that whenever 0 < ''p'' − ''x'' < ''δ'', we have , ''f''(''x'') − ''L'',  < ''ε''.
:$(\forall \varepsilon > 0 )\, (\exists \delta > 0) \, (\forall x \in (a,b)) \, (0 < p - x < \delta \implies , f(x) - L, < \varepsilon)$.

If the limit does not exist, then the oscillation of ''f'' at ''p'' is non-zero.

More general subsets

Apart from open intervals, limits can be defined for functions on arbitrary subsets of R, as follows : let $f : S \to \R$ be a real-valued function defined on arbitrary $S \subseteq \R$. Let ''p'' be a limit point of ''S''—that is, ''p'' is the limit of some sequence of elements of ''S'' distinct from p. Then we say the limit of ''f'', as ''x'' approaches ''p'' from values in ''S'', is ''L'', written
:$\lim_ f(x) = L$
if the following holds:
:For every , there exists a such that for all , implies that .
:$(\forall \varepsilon > 0 )\, (\exists \delta > 0) \,(\forall x \in S)\, (0 < , x - p, < \delta \implies , f(x) - L, < \varepsilon)$.

The condition that ''f'' be defined on ''S'' is that ''S'' be a subset of the domain of ''f''. This generalization includes as special cases limits on an interval, as well as left-handed limits of real-valued functions (e.g., by taking ''S'' to be an open interval of the form $(-\infty,a)$), and right-handed limits (e.g., by taking ''S'' to be an open interval of the form $(a,\infty)$). It also extends the notion of one-sided limits to the included endpoints of (half-)closed intervals, so the square root function ''f''(''x'') = can have limit 0 as x approaches 0 from above:
:$\lim_ \sqrt = 0$
since for every ''ε'' > 0, we may take ''δ'' = ''ε'' such that for all ''x'' ≥ 0, if 0 < , ''x'' − 0, < ''δ'', then , ''f''(''x'') − 0, < ''ε''.

Deleted versus non-deleted limits

The definition of limit given here does not depend on how (or whether) ''f'' is defined at ''p''. 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''. Let $f : S \to \R$ be a real-valued function. The non-deleted limit of ''f'', as ''x'' approaches ''p'', is ''L'' if

:For every ''ε'' > ''0'', there exists a ''δ'' > ''0'' such that for all , implies
:$(\forall \varepsilon > 0 )\, (\exists \delta > 0) \, (\forall x \in S)\, (, x - p, < \delta \implies , f(x) - L, < \varepsilon)$.

The definition is the same, except that the neighborhood now includes the point ''p'', 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) ().

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
:$f(x)=\begin \sin\frac & \text x<1 \\ 0 & \text x=1 \\ \frac& \text x>1 \end$
has no limit at $x_0 = 1$ (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), but has a limit at every other ''x''-coordinate.

The function
:$f(x)=\begin 1 & x \text \\ 0 & x \text \end$
(a.k.a., the Dirichlet function) has no limit at any ''x''-coordinate.

Non-equality of one-sided limits

The function
:$f(x)=\begin 1 & \text x < 0 \\ 2 & \text x \ge 0 \end$
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).

Limits at only one point

The functions
:$f(x)=\begin x & x \text \\ 0 & x \text \end$
and
:$f(x)=\begin , x, & x \text \\ 0 & x \text \end$
both have a limit at ''x'' = 0 and it equals 0.

Limits at countably many points

The function
:$f(x)=\begin \sin x & x \text \\ 1 & x \text \end$
has a limit at any ''x''-coordinate of the form $\frac + 2n\pi$, where ''n'' is any integer.

Limits involving infinity

Limits at infinity

Let $f:S \to\mathbb$ be a function defined on $S\subseteq\mathbb$. The limit of ''f'' as ''x'' approach