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 A B 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 be two inches, and the variable be denoted by x, and the difference between the variable and its limit, by v : Now the sum of the series 1+1+1+1 etc., is evidently 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 variable difference between this variable sum and 2. lim. will be used as an abbreviation for limit. 199. [1] The difference between a variable and its limit is a variable whose limit is zero. [2] If two or more variables, v, v', v', etc., have zero for a limit, their sum, v + v' + v'', etc., will have zero for a limit. [3] If the limit of a variable, v, be zero, the limit of a±v will be the constant a, and the limit of a X v will be zero. [4] The product of a constant and a variable is also a variable, and the limit of the product of a constant and a variable is the product of the constant and the limit of the variable. [5] The sum or product of two variables, both of which are either increasing or decreasing, is also a variable. PROPOSITION I. [6] If two variables be always equal, their limits are equal. Let the two variables AM and A N be always equal, and let A C and A B be their respective limits. A M N P B Since AM approaches indefinitely to C A C, we may suppose that it has reached a value A P greater than A C'. Let AQ be the corresponding value of A N. But both of these equations cannot be true, for A P> A C', and A Q<A B. .. AC cannot be greater than A B. Again, suppose A CAB. Then we may diminish A B to some value A B' such that A C A B'. = Since AN approaches indefinitely to AB we may suppose that it has reached a value A Q greater than A B'. Let A P be the corresponding value of A M. Then Now AP AQ. But both of these equations cannot be true, for A P< A C, and A Q> A B'. .. A C cannot be less than A B. Since AC cannot be greater or less than AB, it must be equal to A B. Q. E. D. [7] COROLLARY 1. If two variables be in a constant ratio, their limits are in the same ratio. For, let x and y be two variables having the constant ratio r, then = y r, or, x=ry, therefore lim. (x) = lim. (r y) = r X lim. (y), therefore [8] COR. 2. Since an incommensurable ratio is the limit of its successive approximate values, two incommensurable ratios a b a' and are equal if they always have the same approximate values b' when expressed within the same measure of precision. PROPOSITION II. [9] The limit of the algebraic sum of two or more variables is the algebraic sum of their limits. Let x, y, z, be variables, a, b, and c, their respective limits, and v, v', and v', the variable differences between x, y, z, and a, b, c, respectively. We are to prove lim. (x+y+2)=a+b+c. Now, x=a-v, y=b—v', z=c-v". Then, x + y + z = a¬v + b − v' + c — v''. a v b с .. lim.(x+y+2)=lim. (a−v+b−v'+c—v''). But, lim. (av + b −v + c¬v") = a+b+c. .. lim. (x + y + 2) = a+b+c. PROPOSITION III. y v' 2 Q. E. D. الوج [6] [3] [10] The limit of the product of two or more variables is the product of their limits. Let x, y, z, be variables, a, b, c, their respective limits, and v, v', v', the variable. differences between x, y, z, and a, b, c, respectively. Now, We are to prove lim. (x y z) = abc. x = a ・v, y = b — v', z = c — v'. Multiply these equations together. Then, x y z = abc terms which contain one or more of the factors v, v', v', and hence have zero for a limit. .. y z) = lim. (x2) 2 lim. (abc #terms whose limits are zero). [8] But lim. (a b c terms whose limits are zero) = a b c. .. lim. (x y z) = a =abc. For decreasing variables the proofs are similar. Q. E. D. NOTE. In the application of the principles of limits, reference to this section (§ 199) will always include the fundamental truth of limits contained in Proposition I.; and it will be left as an exercise for the student to determine in each case what other truths of this section, if any, are included in the reference. PROPOSITION XII. THEOREM. 200. In the same circle, or equal circles, two commensurable arcs have the same ratio as the angles which they subtend at the centre. In the circle APC let the two arcs be AB and AC, and AO B and AOC the & which they subtend. Let HK be a common measure of A B and A C. Suppose HK to be contained in A B three times, At the several points of division on A B and A C draw radii. which AOB will contain three, $ 180 (in the same O, or equal , equal arcs subtend equal at the centre). PROPOSITION XIII. THEOREM. 201. In the same circle, or in equal circles, incommensurable arcs have the same ratio as the angles which they subtend at the centre. In the two equal © ABP and A'B'P' let A B and A'B' be two incommensurable arcs, and C, C' the ▲ which they subtend at the centre. Let A B be divided into any number of equal parts, and let one of these parts be applied to A'B' as often as it will be contained in A'B'. Since A B and A'B' are incommensurable, a certain number of these parts will extend from A' to some point, as D, leaving a remainder D B' less than one of these parts. (two commensurable arcs have the same ratio as the 4 which they subtend at the centre). Now suppose the number of parts into which A B is divided to be continually increased; then the length of each part will become less and less, and the point D will approach nearer and nearer to B', that is, the arc A' D will approach the arc A' B' as its limit, and the Z A'C' D the ZA'C' B' as its limit. |