CHAPTER XXXII.

VARIABLES AND LIMITS.

§ 1. VARIABLES.

1. A Variable is a number that may have a series of different values in the same investigation or problem.

A Constant is a number that has a fixed value in an investi

gation or problem.

Thus, if d be the number of feet a body has fallen from rest in s seconds, it has been shown by experiment that

d = 16 s2.

As the body falls, the distance d and the time s are variables, and 16 is a constant.

Again, time measured from a past date is a variable, while time measured between two fixed dates is a constant.

2. The constants in a mathematical investigation are, as a rule, general numbers, and are represented by the first letters. of the alphabet, a, b, c, etc.; variables are usually represented by the last letters, x, y, z, etc.

3. A variable which has a definite value, or set of values, corresponding to a value of a second variable, is called a Function of the latter.

Thus, 16 x2, ± √(a2 — x2), etc., are functions of x; corresponding to any value of x, the first function has one value, the second has two values.

Again, the area of a circle is a function of its radius; the distance a train runs is a function of the time and speed.

4. Much simplicity is introduced into mathematical investigations by employing special symbols for functions.

The symbol f(x), read function of x, is very commonly used to denote a function of x.

Thus, f(x) may denote x2+1 in one investigation, ax2+be+c

in another.

5. The result of substituting a particular value for the variable in a given expression may be indicated by substituting the same value for the variable in the functional symbol.

Thus, if f(x) = x2+1, then ƒ(a) = a2 + 1, ƒ(2) = 22 + 1 = 5, f(0)=0+1=1.

EXERCISES I.

1. Given ƒ(x) = 5 x2 − 3 x + 2; find ƒ(3), ƒ(0), ƒ(−4), ƒ (x2). 2. Given f(x) = (x − a) (x − b) (x − c); find ƒ(a), ƒ(b).

3. Given ƒ(x) = x2 + 1; find ƒ(x2), [ƒ(x)]2.

4. Given f(x) = x2 − 3 x + 2; find ƒ (x + 4), ƒ (x + h).

--

5. Given f(x) = a2 ; find ƒ (0), ƒ(4), ƒ(− 5), ƒ (x2), ƒ (a).

6. Given ƒ'(x) = x3 + pæ2 + qx + r; find ƒ ( y − ?).

7. Given f(m) = 1 + mx +

m (m 1)

12

find ƒ (5), ƒ (?), ƒ(− 3), ƒ(0).

§ 2. LIMITS.

1. When the difference between a variable and a constant may become and remain less than any assigned positive number, however small, the constant is called the Limit of the variable.

Let the point P move from A towards B (Fig. 4) in the following way: First to P1, one-half of the distance from A to B; next from P1 to P, one-half of the distance from P1 to B; then from P to P, one-half of the distance from P to B; and so on.

Evidently, as P thus moves from A to B, its distance from A becomes more and more nearly equal to AB, and the difference between AP and AB can be made less than any assigned distance, however small, by continuing indefinitely the motion. of P. Therefore AB is the limit of AP.

If we call the distance from A to B unity, we have

Hence,

....

AP1 = }, P1P2 = {, P2Ps= }, P„P1 = {ƒ‚ ··

+++++

....

AP1+ P1P2+ P2P3 + P3P1 + ··· = 1 + 1 + 1 + 16 +····

But, by Ch. XXVII., § 3, Art. 7, the right approaches 1 as a limit.

the sum of the series on That is,

limit of (AP+ P1P2+ P2P3+ P3P1 + ···) = AB.

1

Again, 1+ becomes more and more nearly equal to 1, as

n

n increases indefinitely, and (1+
(1 + 1) − 1,

remain less than any assigned positive number, however small.

2. It follows from the definition of a limit that the variable may be always greater, or always less, or sometimes greater and sometimes less than its limit.

Thus, by Ch. XXVII., § 3, Art. 8, we have

3. The symbol,, read approaches as a limit, or simply approaches, is placed between a variable and its limit.

The word limit may be abbreviated to lim.

lim

Thus, (1x) = 0, read the limit of 1-x, as x approaches

1, is 0.

Infinites and Infinitesimals.

4. The fractions given in Ch. III, § 4, Art. 14 are particular values of the fraction in which the denominator x is as

n

x

sumed to be a variable. It is evident that the value of this fraction can be made greater than any assigned number, however great, by taking its denominator sufficiently small.

A variable which can become and remain numerically greater than any assigned positive number, however great, is called an Infinite Number, or simply an Infinite.

5. The fractions given in Ch. III, § 4, Art. 19, are also particular values of the fraction, in which, as above, the denominator x is assumed to be a variable. It is evident that the value of this fraction can be made less than any assigned number, however small, by taking the denominator sufficiently great.

A variable which can become and remain numerically less than any assigned positive number, however small, is called an Infinitesimal.

No symbol by which to denote an infinitesimal variable has been generally adopted.

It follows from the definition that the limit of an infinitesimal is 0.

6. It is important to keep in mind that both infinites and infinitesimals are variables. Their definitions imply that fixed values cannot be assigned to them.

An infinitesimal should therefore not be confused with 0, which is the constant difference between any two equal numbers.

7. The statement, x becomes infinite, or x increases numerically beyond any assigned positive number, however great, is frequently abbreviated by the expression, x=∞.

8. The conclusions reached in Ch. III, § 4, Arts. 14 and 19, can now be restated thus:

(i.) If the numerator of a fraction remain finite and not 0, and the denominator approach zero, the value of the fraction will become infinite; or stated symbolically,

∞, as x = 0,

wherein n is finite and not 0.

(ii.) If the numerator of a fraction remain finite and not 0, and the denominator become infinite, the value of the fraction will approach 0; or stated symbolically,

n =0, as x = ∞,

wherein n is finite and not 0.

Observe that these principles hold not only when n is a constant, not 0, but also when n is a variable, provided it does not become infinite.

9. The difference between a variable and its limit is evidently an infinitesimal; that is,

if lim xa, then lim (x

Consequently, if lim xa, we have

a) = 0.

x — a = x', or x = a + x',

wherein a' is a variable whose limit is 0.

Conversely, if x = a + x', and a' be a variable whose limit is 0, then lim x = ɑ.

10. If the limit of a variable be 0, the limit of the product of the variable and any finite number is 0. That is,

0, and a be any finite number, lim ax =

if lim x= 0. Let k be any number, however small. Then x can be made less numerically than and, therefore, ax less than k. Hence, lim ax = 0.

k

a

Fundamental Principles of Limits.

11. (i.) If two variables be always equal, and one of them approach a limit, the other approaches the same limit. That is, if x=y, and x = a, then y = a.