Ex. 2. The function (1 — an) a1/", where a > 1, becomes 0 × ∞ when n0. But

and in this form it becomes 0/0 when n = +0.

289. That any quantity whatever may appear as the result in the evaluation of any of the indeterminate forms thus far considered is at once evident from the consideration that the one form 0/0 may be taken as the general representative of the entire set of four forms, and that, a being an arbitrary finite quantity,

a x0 = 0, 0x00, and 0/∞ = 0;

for, from these three equations we derive, by the proper multiplications and divisions,

a = 0/0, 0=0/0, and ∞ = 0/0,

respectively. The multiplications and divisions here performed are, of course, merely symbolic.

The important fact to be remembered, in this connection, is that the indeterminate form gives no information concerning the critical value of the function that gives rise to it.

EXAMPLES LXXXI.

Evaluate the limits of the following functions for the values of x indicated, taking account of positive square roots only.

1. (x2-x-2)/(x2 + x − 6) when x 2.

2. (x3 − x2 − x + 1) / (x3 + x2 − 5x + 3) when x = 1.

3. (xn-an)/(x − a) when x = α.

13. Solve the equation + √(x2 — 9) − (x − 9) = 0. [See page

248.]

14. Solve the equation + √(x2 + 2 ax) − (x + a) = 0.

CHAPTER XXIX.

EXPONENTIALS AND LOGARITHMS.

290. The inverse operations of raising to powers and extracting of roots, considered in Chapter XX., may be regarded as presenting themselves through the medium of the equation

in the two inverse problems:

(1) Given a known number x and a known integer k, to find y. (Involution.)

(2) Given a known number y and a known integer k, to find x. (Evolution.)

In these problems the exponent, as it appears either in x or in y1, is always to be regarded, either as an integer, positive or negative, or as the reciprocal of such an integer; the case where it is a fraction whose numerator is not unity calling for the application of the processes of involution and evolution in succession.

Thus, if the form 63/5 presents itself, we first find 63, or 216, and seek the fifth roots of 216; or, we may reverse the order of this work and get first the fifth roots of 6 and then cube each one of them.

Involution and evolution, therefore, deal primarily with such expressions as

291. If now the form of the expression x be changed to b, for the purpose of indicating that the exponent may become an unknown quantity (the variable), the equation

bx = y

gives rise to the two inverse problems:

*

(1) Given x and b, any known numbers, to find y. (2) Given y and b, any known numbers, to find x. The operation upon b which produces b* is called exponentiation, b is called the base, and b (sometimes also written exp) is called the exponential of a with respect to the base b.

The inverse operation of producing x, when b and y are given, is called logarithmic operation, and is expressed symbolically by the equation

x= 'logy,

in which 'logy is read logarithm of y with respect to the base b.

It is an immediate consequence of this definition that

y = bology.

EXPONENTIATION.

292. The process of exponentiation seeks the value of expressions of the form b2, in which a and b are any real numbers.† But in thus generalizing the combined

* A third case may also present itself; namely, given x and y,to find b. But this is identical with (1), for the equation b = y may be replaced by b = y1/x.

† In the most general interpretation, exponentiation also allows b and x, in the expression be, to have imaginary and complex values.

operations of involution and evolution, it so restricts (by arbitrary definition) the interpretation of its results that to an unique pair of values of a and b there corresponds one and only one value of b, so that while b1, regarded as an example in evolution, has k roots, as the exponential of 1/k to base b it has but one value. The exponential will therefore be so defined as to embody this restriction.

As already indicated, this restriction is arbitrary, but in the chapter on exponential and logarithmic series, a definite algebraic form will be assigned to b, namely, a series whose terms are integral powers of x having coefficients that involve only b and certain numerical fractions, and in this form the uniqueness of the value of ba, for a given value of x, will be apparent.

293. In defining b* for our present purposes we suppose that b is real, positive and greater than 1, and should x be irrational, we replace it by one of the rational fractions m/n, or (m +1)/n,* which, by properly choosing the integers m and n, may be made to differ from x by an arbitrarily small quantity, that is, by a quantity that is smaller than 1/n, where n is as large as we choose to make it. [See Art. 249.] The values of b* we have to

But the discussion of these more general cases does not fall within the scope of this elementary treatise. For a fuller interpretation of exponentiation and logarithmic operation the student is referred to Chrystal's Algebra, Vol. II., Chapter XXIX., and to Stringham's Uniplanar Algebra, Chapter III.

*It must not be inferred that these restrictions are necessary in the general theory of exponentials. [See foot-note to Art. 292.]