CHAPTER XXIII. VARIABLES AND LIMITS. 375. Constants and Variables. A number that, under the conditions of the problem into which it enters, may be made to assume any one of an unlimited number of values is called a variable. A number that, under the conditions of the problem into which it enters, has a fixed value is called a constant. Variables are represented by x, y, z; constants by a, b, c, and by the Arabic numerals. 376. Limits. When the value of a variable, measured at a series of definite intervals, can by continuing the series be made to differ from a given constant by less than any assigned quantity, however small, but cannot be made absolutely equal to the constant, the constant is called the limit of the variable, and the variable is said to approach indefinitely to its limit. Consider the repetend 0.333..., which may be written 1% +180 + TO‰O +..... The value of each fraction after the first is one tenth of the preceding fraction, and by continuing the series we shall reach a fraction less than any assigned value, however small; that is, the values of the successive fractions approach O as a limit. The sum of these fractions will always be less than ; but the more terms we take, the nearer does the sum approach as a limit. Suppose a point to move from A toward B, under the conditions that the first A second it shall move M M' M" B the next one half the distance from A to B, that is, to M; second, one half the remaining distance, that is, to M'; the next second, one half the remaining distance, that is, to M"; and so on indefinitely. Then it is evident that the moving point may approach as near to B as we please, but will never arrive at B. For, however near it may be to B at any instant, the next second it will pass over one half the interval still remaining; it must, therefore, approach nearer to B, since half the interval still remaining is some distance, but will not reach B since half the interval still remaining is not the whole distance. Hence, the distance from A to the moving point is an increasing variable, which indefinitely approaches the constant AB as its limit; and the distance from the moving point to B is a decreasing variable, which indefinitely approaches the constant zero as its limit. If the length of AB is two inches, and the variable is denoted by x, and the difference between the variable and its limit by y: after four seconds, x=1+ } + { + }, y = }; and so on indefinitely. Now the sum of the series 1 + 1 + 1 + 1, etc., is less than 2; but by taking a great number of terms, the sum can be made to differ from 2 by as little as we please. Hence, 2 is the limit of the sum of the series, when the number of the terms is increased indefinitely; and 0 is the limit of the difference between this variable sum and 2. 377. Test for a Limit. In order 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 the constant can be made as near to 0 as we please, but cannot be made absolutely equal to 0. A variable may approach a constant without approaching it as a limit. Thus, in the last example x approaches 3, but not as a limit; for 3 x cannot be made as near to 0 as we please, since it cannot be made less than 1. 378. Infinites. As a variable changes its value, it may constantly increase in numerical value; if the variable can become numerically greater than any assigned value, however great this assigned value may be, the variable is said to increase without limit, or to increase indefinitely. When a variable is conceived to have a value greater than any assigned value, however great, the variable is said to become infinite; such a variable is called an infinite number, or simply an infinite, and is denoted by ∞. 379. Infinitesimals. As a variable changes its value, it may constantly decrease in numerical value; if the variable can become numerically less than any assigned value, however small this assigned value may be, the variable is said to decrease without limit, or to decrease indefinitely. In this case the variable approaches 0 as a limit. When a variable which approaches 0 as a limit is conceived to have a value less than any assigned value, however small this assigned value may be, the variable is said to become infinitesimal; such a variable is called an infinitesimal number, or simply an infinitesimal. 380. Infinites and infinitesimals are variables, not constants. There is no idea of fixed value implied in either an infinite or an infinitesimal. 381. An infinitesimal is not 0. An infinitesimal is a variable arising from the division of a quantity into a constantly increasing number of parts; 0 is a constant arising from taking the difference of two equal quantities. 382. Finite Numbers. infinite is said to be finite. A number which cannot become Theorems of Infinites and Infinitesimals. 383. THEOREM 1. If x is infinitesimal, and a is finite and not 0, then ax is infinitesimal. For ax can be made as small as we please since x can be made as small as we please. 384. THEOREM 2. If X is infinite, and a is finite and not 0, then aX is infinite. For aX can be made as large as we please since X can be made as large as we please. 385. THEOREM 3. If x is infinitesimal, and a is finite and a Χ For can be made as large as we please since x can be x made as small as we please. 386. THEOREM 4. If X is infinite, and a is finite and For a X can be made as small as we please since X can be X made as large as we please. In the above theorems a may be a constant or a variable; the only restriction on the value of a is that it shall not become either infinite or 0. α 0 387. Abbreviated Notation. The expression cannot be interpreted literally, since we cannot divide by 0; neither a can =0 be interpreted literally, since we can find no ∞ number such that the quotient obtained by dividing a by that number is 0. The expression α If α ∞ is simply an abbreviated way of writing: == X, and x approaches 0 as a limit, X increases without limit. The expression α = 0 is simply an abbreviated way of writing: x, and X increases without limit, x approaches 0 as a limit. 388. Approach to a Limit. A variable may approach its limit in one of three ways: 1. The variable may be always less than its limit. 2. The variable may be always greater than its limit. 3. The variable may be sometimes less and sometimes greater than its limit. If a represent the sum of n terms of the series 1 + į + {+} + x is always less than its limit 2. If x represent the sum of n terms of the series 3 x is always greater than its limit 2. If x represent the sum of n terms of the series 3 we have x = 3 3(-)" = 2 − 2 (−)". - (§ 364) As n is indefinitely increased, x evidently approaches 2 as a limit. If n is even, x is less than 2; if n is odd, x is greater than 2. Hence, if n is increased by taking each time one more term, ≈ will be alternately less than and greater than 2. If, for example, |