The method of limits, in its arithmetical formulation, underlies the entire development of the infinitesimal calculus and has important applications in geometry and mechanics. The general principle of convergence to a limit, summed up in the general theorem just stated, provides a criterion for the existence of the limit of a sequence of numbers ; and a considerable part of modern mathematical analysis is devoted to the problem of obtaining special forms of the general principle suited to particular classes of cases. The logical development of the theory depends essen tially on the introduction of irrational numbers and cannot be carried through without a previous development of the theory of these numbers. (See NUMBER and NUMBER SEQUENCES.) Typical Limiting Processes.—If a and v are variables ap proaching the limits a and b respectively and if c is a constant, then the variables cu, u+v, u— v, uv, u/v approach respectively the limits ca, ad-b, a—b, ab, a/b, the last holding only when b is dif ferent from zero. These facts afford the rules for the elementary operations with limits. (See also SERIES.) The proposition that x approaches the limit a is written in the symbolic form lim x=a. If x approaches the limit a then a func tion f(x) of x may approach a limit A; that is, as x varies ac cording to its law of variation, the variation of f(x) according to the induced law of variation may be such that the difference between f(x) and A becomes and remains smaller than any assigned number. In that case we write lim f(x)=A x=a If x varies so as to become numerically larger than any pre assigned number whatever, then x is said to become infinite. We write this statement symbolically in the form lim x=00 . It is not to be understood that there is a number infinity (00 ) which x approaches; the symbol merely says that x becomes infinite in the sense of the definition as given. As x becomes infinite a
function f(x) of x may approach a limit A or it may become infinite; in these respective cases we write lim f (x) =A, lim f (x) =00 x= co x=00 Of course, as x approaches a limit or becomes infinite a given function may neither become infinite nor approach a limit. Thus if x approaches zero as a limit over all numbers near zero the function sin (i/x) does not approach a limit but oscillates be tween the bounds — i and 1.
If a2, a3, . . is a sequence of real numbers and if a num ber a exists such that for every positive number e each of an infinite number of the elements a, . . lies between a—e and a-Fe then a is said to be a value approached by the sequence. In this case a subsequence of the given sequence exists of such sort as to admit the limit a If there is a subsequence of positive numbers in the given sequence such that the elements of the sub sequence become infinite then the given sequence is said to admit 00 as a value approached ; if a subsequence of negative numbers has this property then the given sequence is said to admit —00 as a value approached. The greatest value approached is called by Cauchy the greatest limit of the sequence; the least value ap proached is called the lowest limit of the sequence. If the great est limit and the lowest limit are equal, then the variable has a limit in the sense of the earlier definition. In fact it may be shown that a necessary and sufficient condition that a sequence shall have a limit in the sense of the earlier definition is that it shall admit but one value approached.