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' 1 2n-1 (635) approaches the limit zero as n is indefi nitely 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 given above is 2"-1, which is an infinite variable, if n is allowed to increase indefinitely. 1 2n-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 approaches a limit a, thus, xa. 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 k be any assigned number; then, by hypothesis, e can be made less than! k 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, infinitesimals is an infinitesimal; i. e., if then e=0, e1 = 0, 0, e0,... en =0, e+e+e+...+e=0. of 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 2 and e1e2 = 0. 2 For, let k be any assigned number <1; then e1, e, can each be made less than (8637); hence e, can be made less than k2, which is less than k, since k < 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., lim (er) = As in theorem V, 642, c lim(x). 644. THEOREM VII.-If the sum of a finite number of variables (x1, x2, x) is variable, and if each variable approaches a limit, X2) then the limit of their sum is equal to the sum of their limits; i. e., 2 a finite number of variables (x + x + ) 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 limits; i. e., with the same relations as in 645 it is to be proved Since lim (xx) = a,a,, and . may be considered as a single variable and aa, as its limit, then we have On continuing this mode of reasoning, it follows that the theorem is true for any number of variables. COROLLARY.-If the product of a finite number of variables ... x) is constant, then this constant is equal to the product of their limits; i. e., if 123=c, then, lim lim.c lim 3 = c. The proof is left to the student. 646. THEOREM IX.-If the quotient of two variables, x and y, which approach limits, is a variable, then the limit of their quotient is equal to the quotient of their limits, provided their limits are finite and different from zero; i. e., COROLLARY 1. If the quotient of two variables, x and y, is a constant, c, then c is equal to the quotient of their limits; i. e., if = C. ... [8643] COROLLARY 2. The limit of the quotient of a constant, c, and a variable, x, is equal to the constant, c, divided by the limit of the variable; i. e., For, let and x lim x lim 647. THEOREM X.-The limit of a power of a variable equals the same power of the limit of the variable. where n is positive or negative, integral or fractional. I. When n is a positive integer. |