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Calculus is the study of change. A course in calculus is a gateway to other, more advanced courses in

mathematics devoted to the study of functions and limits, broadly calledmathematical analysis.

In common speech, an infinitesimal object is an object which is smaller than any feasible measurement,

but not zero in size; or, so small that it cannot be distinguished from zero by any available means. Hence,when used as an adjective, "infinitesimal" in the vernacular means "extremely small". In order to give it a

meaning it usually has to be compared to another infinitesimal object in the same context (as in

aderivative). Infinitely many infinitesimals are summed to produce anintegral.

In the 19th century, infinitesimals were replaced bylimits. Limits describe the value of afunctionat a

certain input in terms of its values at nearby input. They capture small-scale behavior, just like

infinitesimals, but use the ordinaryreal number system. In this treatment, calculus is a collection of

techniques for manipulating certain limits. Infinitesimals get replaced by very small numbers, and the

infinitely small behavior of the function is found by taking the limiting behavior for smaller and smaller

numbers. Limits are the easiest way to provide rigorous foundations for calculus, and for this reason they

are the standard approach.

Inmathematics, the limit of a function is a fundamental concept incalculusandanalysisconcerning the

behavior of thatfunctionnear a particularinput.

Formal definitions, first devised in the early 19th century, are given below. Informally, a function fassigns

anoutputf(x) to every inputx. The function has a limit L at an inputp iff(x) is "close" to L wheneverxis

"close" top. In other words, f(x) becomes closer and closer to L asxmoves closer and closer top. More

specifically, when fis applied to each input sufficientlyclose top, the result is an output value that

is arbitrarilyclose to L. If the inputs "close" top are taken to values that are very different,

the limit is said to not exist.

To say that

means that (x) can be made as close as desired to L by makingxclose enough, but not equal, top.

Functions on the real line

Suppose f: RR is defined on thereal lineandp,LR. It is said the limit

offas x approaches p is L and written

if the following property holds:

For every real > 0, there exists a real > 0 such that for all real x, 0 < | xp | < implies

| f(x) L | < .

Note that the value of the limit does not depend on the value of f(p), nor even thatp be in the domain

off.

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A more general definition applies for functions defined onsubsetsof the real line. Let (a, b) be

anopen intervalin R, andp a point of (a, b). Let fbe a real-valued function defined on at least all of

(a, b) \ {p}. It is then said that the limit offasxapproachesp is L if, for every real > 0, there exists a

real > 0such that 0 < |xp | < andx (a, b) implies | f(x) L | < . Note that the limit does not

depend on f(p) being well-defined.

The letters and can be understood as "error" and "distance", and in fact Cauchy used as an

abbreviation for "error" in some of his work (Grabiner 1983). 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.

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