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LIMIT : LIMITS, THEORY OF. 



LIMIT : LIMITS, THEORY OF. 



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the Ukraine, Moldavia, and the neighbouring parts of Europe ; and the 

 choice liqueur rosrtytiu is mainly prepared from it. 



LIMIT : LIMITS, THEORY OF. The word limit implies a fixed 

 magnitude to which another and a variable magnitude may be made as 

 nearly equal as we please, it being impossible however that the variable 

 magnitude can absolutely attain, or be equal to, the fixed magnitude. 

 In this strict sense of the word there are two conditions which must 

 be fulfilled before A can be called the limit of p : first, p must never 

 become equal to A ; secondly, P must be capable of being made as 

 nearly equal to A as we please. 



The method of limits is in reality nothing more than one way of 

 evading the use of the word infinite in an absolute sense [INFINITE] : 

 which may be shown as follows. If we take two common algebraical 

 expressions, such as x and xx, or x 2 , there can be no objection to 

 saying that when x=7, 3^ = 49, because 7 is a definite number, and the 

 operation 7 x 7 is perfectly intelligible. And we may, if we please, 

 say that when x approaches 7, x 1 approaches 49, so that if x may be 

 made as near as we please to 7, ~J? can be made as near as you please to 

 49. Or, 7 being the limit of x, 49 is the limit of x'. The preceding is 

 superfluous, because it is more simple to say at once that x* is 49 when 

 x it 7. But suppose that x, instead of being taken at pleasure, must be 

 determined by means of y ; and let the investigation of the relation 

 between x and y lead to 



then, 80 long as y has any finite value, x must be more than 7 ; nor can 

 the assertion z=7 be made without the implication that y is infinite. 

 In this case then we can only say that x can be made as near as you 

 please to 7, if we may take y as great as we please ; in which case a- 2 

 can be made as near as you please to 49. In the language of the article 

 infinite, we say (for abbreviation, as explained in INFINITE) that xia 7 

 and x' is 49, when y is infinite : iu the language of the present article, 

 we say that x has the limit 7, and x z the limit 49, when y increases 

 itrithout limit. We shall now translate the various illustrations given 

 in the article just cited, from the language of infinites into that of 

 limits. 



When z is infinite, A is equal to B. If A be a fixed magnitude, read 

 If z increase without limit, A is the limit of B : if B be a fixed magni- 

 tude, read If z increase without limit, B is the limit of A : if both A 

 and B be variables, read When z increases without limit, A and B 

 approach to the same limit. 



A finite quantity x, divided by an infinite quantity, is nothing. For 

 this read When the denominator of a fraction increases without limit, 

 the numerator remaining the same, the fraction diminishes without 

 limit. 



Every circle is a regular polygon of an infinite number of sides. For 

 thii read If the number of sides of a regular polygon inscribed in a 

 circle be increased without limit, the polygon approaches without limit 

 to the circle : or, the circle is the limit of all the regular polygons 

 which can be inscribed in it. 



When x is infinite, A and B are both infinite, but A is infinitely 

 greater than B. For this read When x increases without limit, A and 

 B both increase without limit, but the ratio of A to B also increases 

 without limit, or the ratio of B to A diminishes without limit. 



When x = a, z in infinite. For this read When x approaches with- 

 out limit to a, z increases without limit. 



Two infinitely great quantities may have a finite ratio. For this 

 read When two quantities increase without limit, their ratio does not 

 necessarily increase without limit, but may have a finite limit. 



Two infinitely small quantities may have a finite ratio : or when 

 two quantities diminish without limit, their ratio does not necessarily 

 diminish without limit, but may have a finite limit. 



When A is infinitely small, B is infinitely great. For this read 

 When A diminishes without limit, B increases without limit. 



An infinitely small arc of a curve is equal to its chord. For this 

 read When the arc of a curve diminishes without limit, the ratio of 



arc 

 the arc to the chord, or the fraction "^ord , approaches the limit unity. 



Of two infinitely small quantities, one may be infinitely smaller than 

 the other. For this read When two quantities diminish without 

 limit, it is also possible that their ratio may diminish without limit. 



Hitherto we have been dealing with considerations which may appear 

 purely verbal, but which are nevertheless connected with the trans- 

 lation of two very different modes of conception each into the other. 

 But were they declared to be merely verbal, they would not be unim- 

 portant. It is of great consequence that the fundamental notions of 

 mathematics should be expressed in those terms which have always 

 represented the rude and unrigorous form in which they are expressed 

 in common life : and also, when the form just alluded to has given 

 birth to several different modes of expression, it is necessary to point 

 out the connection of these with each other, and to assimilate their 

 defined meanings. But, so far as demonstration is concerned, we have 

 made no step by using one form of words instead of another, or even 

 by substituting the notion of a limit unattainable fur that of the same 

 magnitude attained by the supposition of absolute infinity. The 

 theorem by which rigorous results are obtained is the following : If 



ABTS AND SCI. DIV. VOL. V. 



two variable magnitudes, A and B, be always equal, and if they have 

 limits, namely, p the limit of A, and Q of B ; then p and Q must be 

 equal. This proposition may seem almost self-evident ; it is not 

 however a perfect axiom, and the method of exhaustions [GEOMETRY] 

 was employed by Archimedes to prove it, or rather, to prove the pro- 

 position that if two variable magnitudes be always in a given ratio, 

 their limits are in that ratio. The latter form of the proposition is 

 requisite in Geometry [PROPORTION] ; the former is sufficient in 

 Algebra ; and the proof is as follows : Supposing A and B for instance 

 to be varying lines, always equal, let their limits, if possible, be the 

 unequal lines K L and MN. 



X- 



H 



Since A and B are equal, and since the first can be made 'as near as 

 we please to K L, and the second to M N, it follows that the latter pair 

 are as nearly equal as we please. But this is not true, since the limits 

 are fixed and invariable magnitudes, differing (if they differ at all) by 

 a fixed and invariable quantity. Consequently the limits cannot be 

 other than equal. The proof of the proposition of Archimedes is given 

 in GEOMETRY. 



This proposition, being once understood, is more fruitful in applica- 

 tions than almost any other. We shall give one instance from geometry 

 and one from algebra. 



Circles are to one another as the squares on their diameters. For 

 this proposition is evidently true of the regular polygons inscribed in 

 the two circles with the same number of sides ; and the polygons may 

 be made as nearly equal as we please to the circles. The limits of the 

 polygons then (or the circles themselves) are in that ratio which the 

 polygons always preserve. 



As an instance from algebra, apply the BINOMIAL THEOREM to the 

 development of 



which gives, by an easy transformation, 



1 1^ 1 2 





a series which (by the method in CONVERGENT) is always convergent 

 when nx is less than unity. Apply the same method to the develop 

 nient of 



Z 

 (1 +nx)*= B; 



which gives in the same manner 



l+yx+y-^- x>+y 



x* + , 



(B). 



Now B is evidently A ; and if when n diminishes without limit, B 

 and A approach the limits P and Q, then B and A * (equal quantities) 

 will approach the limits Q and p * , which are therefore equal. But the 

 limit of A, when n diminishes without limit, is 



i T 2.3 

 That of B, on the same supposition, is 

 x- y* 



= p. 



2.3 



+' 



Hence the second of these series is the j/th power of the first; a 

 theorem which the algebraical student will recognise as one of the most 

 important in that science. 



The method of limits generally means the Differential Calculus 

 exhibited upon the principles explained in the article DIFFERENTIAL 

 COEFFICIENT. It is admitted, by a large majority of those who are 

 capable of forming a judgment, that the method by which this theory 

 should be established is either the method of limits, or that of La- 

 grange [FUNCTIONS, THEORY OF], or a mixture of the two. The 

 number of those who contend for the second has very much diminished 

 of late years; and the controversy (if such a thing can be said to 

 exist) lies between the first and third. The reader will find in the 

 eighth number of the Treatise on the subject, published by the 

 Society for the Diffusion of Useful Knowledge, some additional reasons 

 for considering the use of assumed expansions as fallacious. See also 

 SERIES. 



It has been 'customary in elementary mathematical works to en- 

 deavour to postpone the theory of limits as late as possible. Such an 

 attempt can never be very successful ; a clear understanding of the 

 notion of a limit may easily be, and often is, deferred tine die, but the 

 necessity for such an understanding enters with the sixth book of 

 Euclid. We shall even undertake to show [PROPORTION] that the 

 fifth book cannot be properly understood without it. 



Wherever we have occasion to speak of a supposition as true, liyia 

 ever small a quantity may be, we are really involving the notion of a 



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