Limit

LIMIT, a mathematical concept of great importance which has emerged slowly through long historical stages ; it is best pre sented through an account of its origin and development. Among the ancients the method of exhaustion played the role which in more recent times has been taken by the method of limits; the former is ascribed to Eudoxus (q.v.) of Cnidus. It was frequently employed by Euclid (q.v.), and was further developed by Archi medes (q.v.). In one form it may be stated as follows: If from a magnitude more than half of it is taken away, and from the remainder more than half is taken, and from what then remains more than half is again taken, and if this process is continued, the magnitude remaining will ultimately become less than any magnitude which may have been pre-assigned. This principle is frequently employed in the Elements of Euclid for finding areas and volumes. To find the area of a circle, for instance, we may inscribe a regular polygon and find its area; then we may inscribe a larger regular polygon including the former and differing in area from the area of the circle by less than half the difference of area between the preceding polygon and the circle ; we may then repeat the process indefinitely. According to the principle of exhaustion a polygon will thus be obtained whose area differs from that of the circle by less than any pre-assigned magnitude. Nowadays we say that the areas of these inscribed polygons ap proach the area of the circle as a limit in accordance with the f ol lowing abstract definition of limit : A variable x is said to approach a given constant a as a limit if the law of variation of x is such that the numerical value of the difference between x and a becomes and remains smaller than any (whatever) pre-assigned positive number.

In finding the area of a circle by the method just indicated the variable x has for its values the areas of the inscribed poly gons while the constant a is the area of the circle. Here the va riable x remains always less than its limit. If we should similarly find the area a ui a circle by means of circumscribed polygons, then the corresponding variable x would always remain greater than its limit a. According to the definition in its general form x may vary in such a way as to be sometimes less than a and sometimes greater than a and indeed sometimes equal to a. What

is essential is that it shall vary so that the numerical value of the difference between a and x shall become and shall remain less than any (whatever) positive number assigned in advance.

The method employed by Archimedes was more general than the simple method of exhaustion already described. It consisted in enclosing the magnitude to be evaluated between two others which can be brought by a definite process to differ by less than any pre-assigned magnitude. Thus in finding the area of a circle he employed regular inscribed and regular circumscribed poly gons and showed that the number of sides could be taken large enough to render the difference in area between such an inscribed and such a circumscribed polygon less than any pre-assigned magnitude.

Arithmetical Theory of Limits.

The reduction of the idea of limit to the arithmetic form given in the foregoing formal defi nition is due to John Wallis (q.v.). His method, described in mod ern language, rests on the simultaneous consideration of a se quence, a,, a,, . . . of numbers arranged in order according to some law, and a number a such that the varying differences as n increases, become and remain smaller than any pre assigned number whatever. The arithmetic character of the no tion of limit was put in clearer light by Augustin Cauchy (q.v.) in his demonstration of the existence of definite integrals of con tinuous functions. Wallis always supposes essentially that the limit is known or that its existence is evident. In the work of Cauchy we have the clear conception of the existence of the limit as dependent upon certain properties of the numbers con stituting the infinite set of values of the variable. In this con nection the fundamental theorem for limits of sequences is the following : In order that the sequence . . . shall have a limit it is necessary and sufficient that for each positive number E there shall exist a positive integer n such that the difference shall be numerically less than E for every integer m greater than n.