the decimal points altogether, since this amounts to multiplying both numerator and denominator of the complex fraction by the number representing the order of the decimal fractions.

Sections 55-72 may be omitted when the chapter is read for the first time.

55. It frequently happens that when particular values are assigned to the letters appearing in the terms of a fraction, either the denominator or the numerator or both become zero.

For all other finite values of the letters the expressions above assume perfectly definite values.

56. According to definitions previously given, expressions in which zero appears as a divisor have been excluded from calculations as being meaningless as numbers.

In order that such "number forms" may without exception be admitted to our calculations, we proceed to extend our idea of the value of an expression.

Since

Quotient x Divisor Dividend,

we may regard as meaning. Quotient x0 = 0.

Hence, since the product of any number and zero is zero, we may interpret as representing any number.

57. If a variable be supposed to change in value in such a way as to become and remain as nearly equal as we please to some definite fixed value or constant, the variable is said to approach the constant as its limit.

The symbol is read "approaches as a limit."

lim

Xa

E. g. The expression (x + b) = a + b is read, "the limit of (x + b) as approaches a is equal to a + b.”

If we suppose that x approaches 5 as a limit, then so long as x has a value

different from 5,

x 5
Ꮖ 5

will have the value unity (since the numerator and denominator are equal), while x + 5 will differ from 10 by exactly that valne by which x differs from 5.

Accordingly we can make the value of (x+5)*) as nearly equal

to 10 as we please by giving to x a value sufficiently near to 5.

58. We shall define the value of an expression for any particular value of its variable to be the limit (if there be one) ap

proached by the expression as its variable approaches the particular value as a limit.

Although this general definition may be used in all cases, we shall employ it only when, by using the definition given in Chap. I., § 19, we fail to obtain a definite number.

59. To find the value of a fraction which assumes the indeterminate form 0/0 for some particular value of the variable appearing in it, find the limit approached by the fraction when the variable approaches the given particular value as its limit.

60. It should be observed that a rational fraction assumes the form 0/0 because some factor common to both numerator and denominator becomes zero for some particular value of the letter appearing in it.

61. Since the symbol 0/0 does not represent the same value all of the time, but assumes different values according to circumstances, we interpret 0/0 as representing an indeterminate value.

Ex. 1. Find the limiting value of the fraction (x2 − 6 x + 9) / (x − 3) as x 3.

when x is different from 3, is the value of the factor x
As a approaches 3 as a limit the expression x
x
limit.

-

Ꮖ 3'

3.

which is unity

3 approaches zero as a

Accordingly the value of the given fraction is taken as zero when ≈ 3. Another method for finding the limiting value of an indeterminate fraction is shown in the following example:

Ex. 2. Find the value of (2 — 49)/(x + 7) when ≈ ± − 7.

We may indicate that a differs from 7 by writing = 7+h, letting h represent a value which may be made as small as we please.

Accordingly, substituting h7 for x,

2 49
x+7

becomes

-

Hence (x2-49)/(x + 7) differs from 14 by h, which may become and remain as small as we please.

Accordingly the given fraction approaches - 14 as a limit as x approaches

- 7.

62. Consider the fraction ax in which the numerator a is regarded as having some fixed or constant value, different from zero, while the value of the denominator is subject to change.

As the denominator is given successively smaller and smaller values (1/10, 1/100, 1/1000, etc.), the numerator retaining some constant value, it may be seen that as the value of the denominator decreases, the value of the fraction increases.

By giving to the denominator a value small enough, the value of the fraction a/x can be made greater than any assignable number.

63. The symbol, read "infinity", is commonly used to denote all numbers or values which are greater than any assignable arith metic number or value.

The expression = ∞, read x increases in value without limit" or " is infinite", is to be understood as meaning that has no definite fixed value, but that it may assume different values which are very great and are beyond the range of computation or imagination.

64. The symbol for an infinite number,, should never be treated as representing a definite value, and it is not subject to the Laws of Algebra.

+

Corresponding to the ideas of positive and negative numbers, we have to, read "positive infinity," and ∞, read "negative infinity." 65. It is impossible to separate unity, or in fact any number, into such small parts that one of these parts shall have no value at all, that is, be zero. Hence it may be seen that, although the successive denominators (1/10, 1/100, 1/1000, etc.) of the complex fractions in § 62 become smaller and smaller in value, we can never, by diminishing the denominators in this way, obtain zero as a denominator.

66. A variable whose value may become indefinitely small without ever becoming zero is called an infinitesimal.