BOOK VII CHAPTER I LIMITS 634. Constants and Variables.-A constant number is one that always remains the same throughout the investigation. A variable number is one that changes its value, so that at different stages it requires different numerals to express it. In the following pages, the word number will usually be omitted, and the words constant and variable will be used alone. Constants are represented by the first letters of the alphabet a, b, c, . . and by numerals; variables by the last letters of the alphabet x, y, z, 635. Limits.-When a variable takes successive values which approach nearer and nearer to a given constant, so that the difference between the variable and the constant can be made smaller than any assigned number, the constant is called the limit of the variable (345). Suppose that a point moves from O toward X according to the following law; during the first second the point moves one-half the distance from 0 to X and arrives at s; during the second second the point moves one-half the remaining distance s and arrives at s; during the third second one-half the remaining distance s, and arrives at s; and so on indefinitely. FIGURE 1 X Suppose that OX is two feet. Let $1, , 82, 83, etc., be respectively the distances of the point from O, and s, s', s, etc., the distances of the point from X at the end of the first, second, third second and so on, then; after one second s' = 1 + 1 + 1, 1 + 1 + 1 + If the values are represented on a line, it is easy to see the law by 2" 2n-1 which any s can be obtained from its predecessor 8-1, namely; s, lies half way between 8-1 and 2. If therefore n is increased without limit, The same result could formula for the sum s Here have been derived arithmetically from the of the first n terms of the geometric series 'If n increases without limit, 2"-1 approaches 0 as a limit (1635), and 636. Test for a Limit.-The definition of a limit illustrated by the preceding example furnishes a test for a limit; to prove that a variable approaches a constant as a limit, it is necessary and sufficient to prove that the difference between the variable and constant can be made less than any assigned quantity, but can not be made absolutely equal to zero, i. e., their difference approaches the limit 0. 637. Infinitesimals and Infinities. A variable which approaches zero as a limit is an infinitesimal. For example, the difference between a variable and its limit is a variable whose limit is zero. E. g., s,'=2-1 (2635) approaches the limit zero as n is indefinitely increased, and is accordingly an infinitesimal. The reciprocal of an infinitesimal is a variable that can become larger than any assigned quantity and is called an infinite variable. E. g., the reciprocal of the infinitesimal 2-1 given above is 2"-1, which is an infinite variable, if n is allowed to increase indefinitely. 1 REMARK. In all cases, whether a variable actually becomes equal to its limit or not, the important property is that their difference is an infinitesimal. An infinitesimal is not at all times during its existence a very small number. Its virtue lies in the fact that it decreases numerically through positive numbers or increases algebraically through negative numbers, approaching zero as a limit, and not in the smallness of any constant value through which it may pass. FUNDAMENTAL THEOREMS CONCERNING INFINITESIMALS AND LIMITS IN GENERAL 638. THEOREM I.-The product of an infinitesimal e by any finite constant c is an infinitesimal. For brevity we shall express symbolically the fact that a variable x approaches a limit a, thus, x = a. Since e is an infinitesimal, then by definition (8637). e = 0, and similarly for any other infinitesimal. The theorem requires us to prove that if then e = 0, ce 0. For, let be any assigned number; then, by hypothesis, e can be k made less than, i. e., ce can be made less than any assigned number, k, and is, therefore, infinitesimal. 639. THEOREM II.--The algebraic sum of a finite number, n, of infinitesimals is an infinitesimal; i. e., if For, the sum of n variables does not numerically exceed the product of n by the largest of these; but their product by theorem I is an infinitesimal; therefore the sum of the n infinitesimals is an infinitesimal. 640. THEOREM III.-The product of two infinitesimals is an infinitesimal; i. e., if then G = 0 2 = 0, For, let be any assigned number <1; then e1, 2 than 2, which is less than (637); hence e, can be made less less than k, since <1; that is e, can be made less than any assigned number, and is, therefore, infinitesimal. 641. THEOREM IV.-If two variables, x and y, are continually equal and if one of them, x, approaches a limit, a, then the other approaches the same limit; i. e., if Since the difference between a variable and its limit is an infinitesimal (637), then 642. THEOREM V.. The limit of the sum of a constant, c, and a variable, x, equals the sum of the constant and the limit of the variable; i. e., 643. THEOREM VI.-The limit of the product of a constant, c, and a variable, x, is equal to the product of the constant by the limit, a, of the variable; i. e., 644. THEOREM VII.—If the sum of a finite number of variables (x, x, x) is variable, and if each variable approaches a limit, then the limit of their sum is equal to the sum of their limits; i. e., COROLLARY.—If the sum of = [639] lim, + . . + lim x„. a finite number of variables (x,+£2 + r) is constant and if each variable approaches a limit, then this constant, c, is equal to the sum of their limits; i. e., if 645. THEOREM VIII.-If the product of a finite number of variables (x, x, . . . x) is variable and if each variable approaches a Limit, then the limit of their product is equal to the product of their |