Since when v1 and v2 are near their common limit 0, their product is much less than either v1 or v2, Similarly, the principle may be established for any number of variables. 425. PRINCIPLE 7. The limit of the variable quotient of two variables is equal to the quotient of their limits, provided the limit of the divisor is not 0. The above principle may be established as follows: The principle has no meaning when z = 0, since lim. y cannot be divided by 0. 426. When by causing a variable x to approach sufficiently near to a it is possible to make the value of a given function of x approach as near as we please to a finite constant 1, 7 is called the limit of the function when xa. Suppose that .1, .11, .111, .1111, ... are successive values of x approaching as a limit. Then, the corresponding values of 1 – 2x, a function of x, are .8, .78, .778, .7778, values of a variable approaching as a limit. As x = }, the function of x, 1 − 2x, = }; for by causing xr to approach sufficiently near to it is possible to make 1 – 2x approach as near as we please to 7. The expression lim. [function of x],÷a is read limit of function of x as a approaches a as a limit.' 427. In finding the limiting values of the functions given in the following examples, the student is expected to apply the principles that have been established above. Finding the limiting value of a function of x as xa is called evaluating the function for x = a. EXAMPLES If xa, y = 2, and z = 0, find the limit of Since a denotes any number whatever, 0 are symbols for indeterminate numbers. If k is any constant, ∞, + k -- is a symbol for an indeterminate number. ADV. ALG.-25 429. Since every function of a single variable is a variable, it is evident that the preceding principles apply to functions of a variable. Thus, to apply to functions of a variable, Prin. 5, 6, and 7 may be stated as follows: The limit of the sum of a finite number of functions of x is equal to the sum of their limits. The limit of the product of a finite number of functions of x is equal to the product of their limits. The limit of the quotient of two functions of x is equal to the quotient of their limits, provided the limit of the divisor is not zero. These principles fail to give a limit whenever the result obtained involves one of the indeterminate forms, ∞ — ∞, 0 × ∞, ∞ × 0, 0, ∞ 430. The preceding principles of limits lead to the conclusion that the limit of a function is found by substituting the limits of the variables for the variables, except when such a substitution gives an indeterminate form (§ 429). Thus, if lim. x = 5 and lim. y = 2, the limit of 4 x tuting 5 for x and 2 for y in the function 4x employed directly to evaluate the functions - 3y. 3 y is found by substiBut if substitution is 1 1 and a 1 x(x-1) y (x2 − 1) ÷ (x 1) when x1 and y = 0, these functions take the forms ∞ - ∞, ∞ × 0, and 0 ÷ 0, respectively. When the method of evaluation by substitution in the given function fails, the evaluation of the function is performed by the aid of Prin. 2. Thus, to evaluate (x2 - 1)÷(x − 1) when x1, find another function of x, as x + 1, equal to the given function (x2 − 1) ÷ (x − 1) for all values assumed by x while approaching the limit 1. If x takes the successive values 2, 1, 1, 1, 1, 1, ..., approaching 1, then both functions (x2 - 1)÷ (x − 1) and x + 1, take the successive values Since the two functions are equal for all values of x as x approaches its limit 1, by Prin. 2 they have the same limit. This limit is lim. (x + 1).±1, which by substituting lim. x for x is found to be 1 + 1, or 2. +63 x=1 when x=0 SOLUTION. As x approaches the limit 0, the first three terms of the numerator and also of the denominator become infinitely small as compared with the fourth, and, consequently, may be neglected. Hence, when x = 0, the fraction approaches the limiting value §. Asx, that is, as x becomes indefinitely greater, the last three terms of the numerator and also of the denominator become infinitely small as compared with the first, and, consequently, may be neglected. ∞, the fraction approaches the limiting value or 4 4 x8 Hence, when Find the limiting values of the following when a 0 and when INCOMMENSURABLE NUMBERS 431. Though an incommensurable number (§ 227) cannot be expressed by any integer or by any fraction with integral terms, a commensurable number can always be found to differ from any incommensurable number by less than any number that may be assigned, however small. For example, though V2 cannot be expressed by a decimal that terminates, commensurable numbers can be found to approximate to the true value of √2. For by the process of evolution a commensurable number can be found to differ from √2 by less, successively, than .1, .01, .001, .0001, ..., the difference finally becoming smaller than any number, however small, that can be assigned. To generalize let p and q be variables each of which may take any integral value whatever. Then, any commensurable number may be exactly represented by, and since q may be made as q large as we please and p may be given any integral value whatР ever, any incommensurable number may be represented by to q any required degree of approximation. Thus, any incommensurable number may be included between Р 1 and each of which differs from it by less than and P+1 զ զ q since may be made as large as we please, the limit of this difference is zero. Hence, an incommensurable number may be regarded as the limit of a variable commensurable number. Thus, 2 is the limit of the series 1 ++ro + robo + ..., the sum of n terms of which is 1, 1.4, 1.41, 1.414, ..., as n takes the successive values 1, 2, 3, 4, |