Limit

Consider an arbitrary function, y = f(x). (A function is a set of ordered pairs for which the first and second elements of each pair are related to one another in a fixed way. When the elements of the ordered pairs are real numbers, the relationship is usually expressed in the form of an equation.) Suppose that successive values of x are chosen to match those of a converging sequence such as the sequence S from the previous example. The question arises as to what the values of the function do, that is, what happens to successive values of y. In fact, whenever the values of x form a sequence, the values f(x) also form a sequence. If this sequence is a converging sequence then the limit of that sequence is called the limit of the function. More generally when the value of a function f(x) approaches a definite value L as the independent variable x gets close to a real number p then L is called the limit of the function. This is written formally as:

and reads "The limit of f of x, as x approaches p, equals L." It does not depend on what particular sequence of numbers is chosen to represent x; it is only necessary that the sequence converge to a limit. The limit may depend on whether the sequence is increasing or decreasing. That is the limit, as x approaches p from above may be different from the limit as x approaches p from below. In some cases one or the other of these limits may even fail to exist. In any case since the value of x is approaching the finite value p the difference (p-x) is approaching zero. It is this definition of limit that provides a foundation for development of the derivative and the integral in calculus.

There is a second type of functional limit: the limit as the value of the independent variable approaches infinity. While a sequence that approaches infinity is said to diverge, there are cases for which applying the defining rule of a function to a diverging sequence results in creation of a converging sequence. The function defined by the equation y = 1/x is such a function. If a finite limit exists for the function when the independent variable approaches infinity it is written formally as:

and reads "The limit of f of x, as x approaches infinity, equals L." It is interesting to note that the function defined by y = 1/x has no limit when x approaches 0 but has the limit L = 0 when x approaches ∞.