Limit of a function
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.
For the mathematical concept in general, see Limit (mathematics).
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, the output value can be made arbitrarily close to L if the input to f is taken sufficiently close to p. 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[edit]
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 (see (ε, δ)-definition of limit below) to define continuous functions. However, his work was not known during his lifetime.[1]
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 x necessarily produces an infinitesimal change in y, while Grabiner claims that he used a rigorous epsilon-delta definition in proofs.[2] In 1861, Weierstrass first introduced the epsilon-delta definition of limit in the form it is usually written today.[3] He also introduced the notations and [4]
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.[5]
a neighborhood of −∞ is defined to contain an [−∞, c) for some
interval
a neighborhood of ∞ is defined to contain an interval (c, ∞] where and
a neighborhood of is defined in the normal way metric space
Functions of more than one variable[edit]
Ordinary limits[edit]
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 defined on we defined the limit as follows: the limit of f as (x, y) approaches (p, q) is L, written
if the following condition holds:
Other characterizations[edit]
In terms of sequences[edit]
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:
if, and only if, for all sequences xn (with xn not equal to a for all n) converging to a the sequence f(xn) converges to L. It was shown by Sierpiński in 1916 that proving the equivalence of this definition and the definition above, requires and is equivalent to a weak form of the axiom of choice. Note that defining what it means for a sequence xn to converge to a requires the epsilon, delta method.
Similarly as it was the case of Weierstrass's definition, a more general Heine definition applies to functions defined on subsets of the real line. Let f be a real-valued function with the domain Dm(f ). Let a be the limit of a sequence of elements of Dm(f ) \ {a}. Then the limit (in this sense) of f is L as x approaches p
if for every sequence xn ∈ Dm(f ) \ {a} (so that for all n, xn is not equal to a) that converges to a, the sequence f(xn) converges to L. This is the same as the definition of a sequential limit in the preceding section obtained by regarding the subset Dm(f ) of as a metric space with the induced metric.
In non-standard calculus[edit]
In non-standard calculus the limit of a function is defined by: if and only if for all is infinitesimal whenever x − a is infinitesimal. Here are the hyperreal numbers and f* is the natural extension of f to the non-standard real numbers. Keisler proved that such a hyperreal definition of limit reduces the quantifier complexity by two quantifiers.[23] On the other hand, Hrbacek writes that for the definitions to be valid for all hyperreal numbers they must implicitly be grounded in the ε-δ method, and claims that, from the pedagogical point of view, the hope that non-standard calculus could be done without ε-δ methods cannot be realized in full.[24] Bŀaszczyk et al. detail the usefulness of microcontinuity in developing a transparent definition of uniform continuity, and characterize Hrbacek's criticism as a "dubious lament".[25]
In terms of nearness[edit]
At the 1908 international congress of mathematics F. Riesz introduced an alternate way defining limits and continuity in concept called "nearness".[26] A point x is defined to be near a set if for every r > 0 there is a point a ∈ A so that |x − a| < r. In this setting the if and only if for all L is near f(A) whenever a is near A. Here f(A) is the set This definition can also be extended to metric and topological spaces.
