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a=1, and we shall have

16

1

1 1

1 -α 1-1

=2, which will also be equal to the following series, 1+1+! +++++, &c., to infinity (Art. 164). Now, if we take only two terms of the series, we shall have 1+, and it wants of being equal to 2; if we take three terms, it wants, for the sum is 12; if we take four terms, we have 17, and the deficiency is only: Therefore, we see very clearly that the more terms of the quotient we take, the less the difference becomes; and that, consequently, if we continue to take successive portions of this series, the differences between these consecutive

1

sums and the fraction =2, decrease, and end 1-1 by becoming less than any given number, however small it may be. The number 2 is therefore still a limit, according to the acceptation of this word.

Now, it may be observed, that if the preceding series be continued to infinity, there will be no difference at all between its sum and the value of the 1 fraction or 2.

170. Alimit, according to the notion of the ancients, is some fixed quantity, to which another of variable magnitude can never become equal, though, in the course of its variation, it may approach nearer to it than any difference that can be assigned; always supposing that the change, which the variable quantity undergoes, is one of continued increase, or continued diminution. Such, for example, is the area of a circle, with regard to the areas of the circumscribed and inscribed polygons; for, by increasing the number of sides of these figures, their difference may be made

less than any assigned area, however small; and since the circle is necessarily less than the first, and greater than the second, it must differ from either of them by a quantity less than that by which they differ from each other. The circle will thus answer all the 'conditions of a limit, which is included in the above definition.

171. The preceding considerations are very proper to define the nature of the word limit; but as Algebra, which is the subject we are treating of here, needs no foreign aid to demonstrate its principles, it is necessary, therefore, to explain the nature of the word limit, by the consideration of algebraic expressions. For this pur

pose, let, in the first place, the very simple fraction

ax

be in which we suppose that a may be posix+a'.

tive, and augmented indefinitely; in dividing both

a

terms of this fraction by x, the result, evidently

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shows that the function remains always less than a, but that it approaches continually to a, since that the part, of its denominator, diminishes more and more, and can be reduced to such a degree of smallness as we would wish.

The difference between a and the proposed frac

tion being in general expressed by a-

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a2

x+a x+a' becomes so much smaller, according as x is larger, and can be rendered less than any given magnitude, however small it may be; so that the proposed fraction can approach to a as near as we would wish: a is therefore the limit of the fraction

ax

x+a

relatively to the indefinite augmentation which

čan receive. It is in the characters which we have just expressed, that the true acceptation, which we must give to the word limit, consists, in order to comprehend every thing which can relate to it.

If we had remarked in the preceding example, that by carrying on, as far as we would wish, the augmentation of x, we could never regard, as no

a2

thing, the fraction ; therefore we would rea

x+a

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x+a

sonably conclude, that the fraction though it would approach indefinitely to the limit a, could never attain a, and, consequently, cannot surpass it; but it would be wrong to insert this circumstance as a condition in the general definition of the word limit: we would thereby exclude the ratios of vanishing quantities, ratios whose existence is incontestable, and from which we derive much in analysis.

172. In fact, when we compare the functions a and ax+x2, we find that their ratio, reduced to its

a

most simple expression, is a+ and that it approaches nearer and nearer to unity, according as diminishes. It becomes exactly 1, when x=0; but the quantities ax and ax+x2, which are then rigorously nothing, can they have a determinate ratio? This is what appears difficult to conceive; and we cannot give a clear idea of it but by presenting the quantity 1 as a limit to which the ratio of the functions ax and ax+x2 can approach as near

a

as we would wish, since the difference, 1- a + x

00

can be rendered less than any assignable maga+x' nitude, however small this magnitude may be.

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ties ax and ax+x2 can not only attain unity when we make x=0, but surpass it when we suppose a

negative, since it becomes then

α

a-x

a quantity which is greater than 1, when x<a. This circumstance appears not at all contrary to the idea of limit; for we can regard the value 1, which answers to x=0, as a term towards which the ratio of the functions ax and ax+x2 tends, by the diminutions of the values of x, whether positive or negative. For further illustrations of the word limit, and what is meant by infinity, and infinitely small quantities or infinitessimals, the intelligent reader is referred to LACROIX's Introduction to the Traité du Calcul Differentiel et du Calcul Integral, 4to.. where these subjects are clearly elucidated.

173. Now, let a=1, in the fraction

shall have

1

1-3

1

1. α

and we

=2=1+3+}+&+++,&c.

27

If we take two terms, we find 1+1, and the difference; three terms give 1+, the error = for four terms the error is no more than. Since, therefore, the error always becomes three times less, it tends toward zero, which it cannot attain, and the sum tends toward, which is the limit.,

1

Je

174. Again, let us take a=, and we shall have

32 343

= 3 = 1 + 2 + ÷ +&+}{++, &c.; here, in 1-3 the first place, the sum of two terms, which is 1+3. is less than 3 by 1+1; taking three terms, which make 2, the error is ; for four terms, whose sum is 211, the error is 1.

1

175. Finally, for a=1, we find =1+1=1

16 64

1 -1

titätát2+, &c.; the two first terms are equal to 11, which gives for the error; and taking one term more, we shall only have an error of 1.

176. From the preceding considerations we may readily conclude, that any fraction having a compound denominator may be converted into an infinite series by the following rule; and if the denominator be a simple quantity it may be divided into two or more parts.

RULE.

177. Divide the numerator by the denominator, as in the division of integral quantities, and the operation, continued as far as may be thought necessary, will give the series required.

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