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INFINITESIMALS 
 AND LIMITS. 
 
 BY 
 
 JOSEPH JOHNSTON HARDY, 
 
 PROFESSOR IN LAFAYETTE COLLEGE. 
 
 EASTON. PA. : 
 THE CHEMICAL PUBLISHING CO. 
 
 1912. 
 
INFINITESIMALS 
 AND LIMITS. 
 
 BY 
 
 JOSEPH JOHNSTON HARDY, 
 
 PROFESSOR IN LAFAYETTE COLLEGE. 
 
 EASTON. PA. : 
 
 THE CHEMICAL PUBLISHING CO. 
 
 1912. 
 

 Copyright, 1900, by Edward Hart. 
 
 • • .• • • 
 
 • • * • • • 
 
 • • • •• 
 
:3S 
 
 INFINITESIMALS AND LIMITS 
 
 CHAPTER I 
 INFINITESIMALS 
 
 /='' 
 
 ^o 
 
 Fig. I. 
 
 1. In Fig. I let AB and CD be two parallel straight 
 lines and draw EF J_ CD. I,et a point P start from E 
 and move along AB to the right so as at the end of the first 
 second to be at P, at the end of the second second to be 
 at P', at the end of the third second to be at P", 
 
 As the point P changes its position, 
 Z. EFP takes the different values EFP, EFP', EFP", etc. 
 Z PFD '' " PFD, P'FD, P"FD, etc. 
 
 EP '' '' EP, EP', EP", etc. 
 
 FP •' '* FP, FF, FP", etc. 
 
 but ^^ EFD does not change its value, 
 and EF 
 
 That is, among the quantities which enter into the dis- 
 cussion of the given figure as the point P changes its posi- 
 tion, there are some, Z. EFD and EF, which retain the 
 same values, and which are accordingly called constants. 
 
 4 
 
 'S r> /» ^'^ *> 
 
4 INFINITESIMALS AND I^IMITS 
 
 Others, as EP, Z! EFP, ^ PFD and FP, which assume 
 different values, are called variables. . 
 
 /} — — , , , — > a 
 
 Fig. 2. 
 
 Again, suppose a man starts from A to go to B, going 
 over half the distance between himself and B every hour. 
 At the end of the first hour he will reach M ; at the end 
 of the second he will reach M' ; at the end of the third at 
 M"; etc. 
 
 In this illustration we see that AM, the distance the 
 man travels, and MB, the distance from the man to B, 
 change continually ; but no matter how long the man 
 travels, AB always retains the same value. Hence we 
 call AM and MB variables and AB a constant. 
 
 In the series 
 
 I 4- >^ + ^ -f >^ + etc., 
 let n be the number of terms taken beginning with the 
 first and let s be the sum of those terms. 
 
 Now let n be made greater and greater continually. 
 Then when «=2, s=i -{- }4 = i}4 
 
 «=4, s=i-^ y2 -h }i -\- 'A =^V^ 
 
 We see from these equations that n and s change their 
 values continually but that each term of the series always 
 retains the same value. 
 
 Hence we call n and s variables and each term a 
 constant. 
 
 Thus in any problem there may appear quantities 
 which, under the assumptions made in the problem, 
 always retain the same values. There may also appear 
 
INFINITESIMALS AND LIMITS 5 
 
 Others which under the same assumptions take different 
 values. The former are called constants, the latter 
 variables. 
 
 2. A variable is a quantity which changes its value 
 under the assumptions made in the problem into which it 
 enters. 
 
 3. A constant is a quantity which does not change its 
 value under the assumptions made in the problem into 
 which it enters. 
 
 4. The absolute value of a number is simply the num- 
 ber of units it contains, no regard being paid to its sign. 
 
 Thus the absolute value — 5 and of 5 is 5. 
 
 5. An infinitesimal is a variable which approaches zero 
 in such a way that its absolute value may be made to 
 become and remain less than any positive number c that 
 may be named, however small the e may be taken. 
 
 In Fig. 3 let the point P' remain fixed, let P move con- 
 tinually along the curve till it reaches P', and let it stop 
 there. 
 
 Then the ^ c continually approaches zero in such a way 
 
6 INFINITESIMALS AND LIMITS 
 
 that its absolute value may be made to become and 
 remain less than any positive number c however small e 
 may be taken. 
 
 For let € = 1°. 
 
 Then by moving P far enough we can make 
 
 c< 1°. 
 Now let € = i'. 
 
 By moving P farther on we can make 
 
 e< i'. 
 Let e = i". 
 
 Again by moving P farther on we can make 
 
 c < i". 
 
 Similarly no matter how small € is taken we can make 
 
 C<e 
 by moving P on far enough. 
 
 Since by hypothesis P continually approaches P' and 
 does not pass it, c continually approaches o but cannot 
 become less than o. Therefore when the absolute value 
 of c once becomes less than e it must remain so. 
 
 Hence by the definition c is an infinitesimal. 
 
 Again in Fig. 3 the line PK continually approaches 
 zero in such a way that its absolute value may be made 
 to become and remain less than any positive number e, 
 however small the e may be made. 
 
 Let €=.i. 
 
 Then by moving P towards P' we can make 
 
 PK < .1. 
 
 Now let € = .01. 
 
 Then by moving P further on we can make 
 PK < .01. 
 
INFINITESIMALS AND LIMITS ^ 
 
 Similarly, no matter how small e may be taken we can 
 make PK < e 
 
 by moving P far enough. 
 
 Since by hypothesis P continually approaches P' and 
 cannot pass it the absolute value of PK continually 
 approaches o and cannot become less than o. Therefore 
 when the absolute value of PK once becomes less than 
 € it must remain so. 
 
 Hence by the definition PK is an infinitesimal. 
 
 Similarly it may be shown that the A PKP' is an 
 infinitesimal. 
 
 It is obvious that each of these infinitesimals c, PK 
 and A PKP' will become zero when P reaches P'. 
 
 In Fig. 4 let AB and CD be two parallel straight lines. 
 Draw P"K J_ CD. 
 
 C 
 
 Fig. 4. 
 
 Now let B remain fixed and let P move continually 
 along AB towards the right. 
 
 The Zl PHD continually approaches zero and its abso- 
 lute value may be made to become and remain less than 
 any positive number e, how^ever small e may be made. 
 
 Hence by the definition ^ PED is an infinitesimal. 
 
 It is obvious that Zl PED can never be made equal to 
 zero. 
 
8 INFINITESIMALS AND LIMITS 
 
 IvCt 5 = the sum of 72 terms of the series 
 I -V % ^Va -^ }i^ etc. 
 I.et R = 2 — s 
 
 Now let n increase continually. Then s will increase 
 continually and can be made as nearly equal to 2 as we 
 please. Hence R will continually approach zero and 
 may be made less than a positive number e, however 
 small c may be taken. 
 
 Since all the terms are positive, R must be positive ; 
 hence when once R becomes less than c, it must remain 
 so. 
 
 Hence by the definition R is an infinitesimal. 
 
 It is obvious that R can never be made equal to zero. 
 
 6. There are therefore two classes of infinitesimals, 
 namely : 
 
 ( 1 ) Those which may finally become zero. 
 
 (2) Those which can never become zero. 
 
 CHAPTER II 
 INFINITES 
 
 7. An infinite. — An infinite is a variable whose abso- 
 lute value can be made to become and remain larger than 
 any positive number that may be assigned. 
 
 In Fig. 4 the line P"P is an infinite ; for by moving P 
 farther and farther to the right we can make the absolute 
 value of P"P greater than any positive number that may 
 be taken. Also since the absolute value of P"P is 
 always increasing, when once it becomes greater than any 
 positive number, it will remain greater. 
 
. INFINITESIMAI.S AND I.IMITS 9 
 
 Again let 
 
 [i] 5=i-i-2 + 44- etc. to n terms. 
 
 Then 5 is an infinite. 
 
 For the series is a Geometrical Progression and we 
 
 know by arithmetic that 
 
 ar— a 
 [2] s = 
 
 2" — I 
 [3] .-. ^=- ^ = 2" — I. 
 
 L^J 2—1 
 
 Now we see from the right-hand side of [3] that by 
 making n larger and larger we can make s larger than 
 any positive number that may be taken. Also since s 
 is always increasing when once it becomes larger than 
 any positive number, it will remain larger. 
 
 The symbol for an infinite is 00 . 
 
 8. A finite. — Any quantity which is neither an infini- 
 tesimal nor an infinite is called a finite. 
 
 CHAPTER III 
 
 PROPERTIES OF INFINITESIMALS 
 
 9. In Fig. 5 draw PQ, RS, etc., J_ xx' and at equal 
 distances from each other. Also draw RT, etc. , parallel 
 
 to xx\ . , 
 
 We will in this way construct the triangular figure 
 PRT bounded by PT, RT and the curve ; also the trian- 
 gular figures a, b, c, and d ; and the rectangles q, p, and o. 
 
 Now let PQ move continually towards P'Q', the other 
 perpendiculars moving so as always to preserve the 
 
lO 
 
 INFINITESIMALS AND LIMITS 
 
 same relative position. Then it is obvious that PP'Q'Q. 
 ^, /, o, a, b, c, d are all infinitesimals. by § 5. 
 
 Fig. 5- 
 
 We see also that q <p <o <^ PP'Q'Q. 
 
 Hence one infinitesimal may be greater than another. 
 
 Also since 
 a ^ b -^ c-^d -^ etc. -V q ^ p ^ ^ etc. = PP'Q'Q. 
 it is obvious that the sum of a finite number of infinitesi- 
 mals may be an infinitesimal. 
 
 Finally let there be any n -j- i infinitesimals like q, p,o. 
 Then since q <i P <^ ... 
 
 p-\- 0^- ... 
 
 we get 
 Let 
 Then 
 
 q < 
 p-ho 
 
 . = J 
 
 ?< 
 
 But q and 5 are both infinitesimals. 
 
INFINITESIMAI.S AND I.IMITS II 
 
 Hence an infinitesimal may be less than another 
 infinitesimal divided by a constant. 
 
 It is obvious from this discussion that an infinitesimal 
 can be properly said to be very small only at certain 
 stages of its variation. 
 
 When a variable becomes smaller and smaller in such 
 a way that it may be made smaller than any positive num- 
 ber that may be named, it is said to decrease indefinitely. 
 
 When a variable becomes larger and larger in such a 
 way that it may be made larger than any positive number 
 that may be named, it is said to increase indefinitely. 
 
 PROPOSITION I 
 
 10. The reciprocal of an infinitesimal is an infinite. 
 For let a ^ any infinitesimal. 
 
 Then — = its reciprocal, 
 
 a 
 
 Now when a decreases indefinitely, 
 
 — increases indefinitely ; 
 
 a 
 
 for when the denominator of a fraction decreases indefi- 
 nitely the value of the fraction increases indefinitely. 
 
 Hence — is an infinite. by § 7. 
 
 a 
 
 Q. E. D. 
 
 PROPOSITION 2 
 
 11. The product of a constant a nd an infinitesimal is itself 
 an infinitesimal. 
 
 Let € and s = two infinitesimals 
 
 and n = any constant. 
 
 Then we may take 5 < — . by § 5. 
 
 n 
 
 Hence ns < c Q. K. D. 
 
12 INFINITESIMALS AND LIMITS 
 
 PROPOSITION 3 
 
 12. The sum of a7iy finite number of infinitesimals is 
 also an infinitesimal . 
 
 Let there be n infinitesimals. 
 
 Let c^ be that one of these infinitesimals which has at 
 any instant the greatest absolute value. 
 
 [i] Let J = €, -i- e, + €3 + . . . H- 6„. 
 
 [2] Then s < ne^, 
 
 But nt^ is an infinitesimal. by § 11. 
 
 Hence by [2] ^ is also an infinitesimal. q. e. d. 
 
 PROPOSITION 4 
 
 13. The product of any finite number of infinitesimals is 
 also an infinitesimal. 
 
 Let there be n infinitesimals 
 
 ^1. €2. «3. • ••. €„. 
 
 Let ;r be a variable which can be made to increase in- 
 definitely. We can alwa3^s find a value of x such that 
 
 by § 5. 
 
 [i] Hence 
 
 ^I< 
 
 I 
 X ' 
 
 
 e.< 
 
 I 
 X ' 
 
 
 ^3< 
 
 I 
 
 X ' 
 
 
 etc < etc., 
 
 
 ^«< 
 
 I 
 
 X ' 
 
 
 Cj €, €3 .... 
 
 €„< 
 
 I 
 x„ 
 
INFINITESIMAI.S AND LIMITS 1 3 
 
 Now making x increase indefinitely we can make — an 
 
 infinitesimal. Hence by [i] c^ f..^ S • • • ^n is an infin- 
 itesimal. Q. E. D. 
 PROPOSITION 5 
 
 14. First. — A power of an infinitesimal is itself an 
 infinitesimal if its exp orient is a positive finite other than o. 
 
 Second. — // is an infinite if its exponent is a negative 
 finite other than o. 
 
 Third. — // is equal to i if its exponent is o. 
 
 First. 
 
 Let a be any infinitesimal and p a positive finite other 
 than o. 
 
 [i] Then a^ = a a a a ... to / factors. 
 
 [2] But a a a ... to / factors is an infinitesi- 
 
 mal, by § 13. 
 
 [3] Hence a^ ^^ an infinitesimal. 
 
 Second. 
 Let — / be a negative finite other than o. 
 
 [4] Then ''~'=^' 
 
 But since a is an infinitesimal 
 
 [5] a^ = an infinitesimal, by case i. 
 
 by § 10. 
 
 [6] and 
 
 
 I 
 
 = an 
 
 infinite. 
 
 Hence by [4] 
 
 we 
 
 get 
 
 
 
 [7] 
 
 
 a^ 
 
 — an 
 Third. 
 
 infinite. 
 
 Let 
 
 
 P 
 
 = 0. 
 
 
 Then 
 
 
 a.' . 
 
 = a« = 
 
 = I. 
 
 Q. E. D. 
 
H 
 
 INFINITESIMALS AND LIMITS 
 
 CHAPTER IV 
 
 LIMITS 
 15. Limit. — The limit of a variable is that constant 
 which differs from the variable by an infinitesimal. 
 
 ;■• • ' 
 
 
 x:^ 
 
 <- 
 
 
 f. 
 
 
 rt 
 
 
 _yyi 
 
 
 
 -n 
 
 r 
 
 
 ^ 
 
 / 
 
 
 
 ti 
 
 Fig. 6. 
 
 h 
 
 In Fig. 6 
 
 [i] ^^a= Z.b -\- Z.C by Geom. 
 
 [2] .-. ,^a — Z.b = ^^c 
 
 Now let P move along the curve till it reaches P'. 
 
 Then ^^i is a variable, z^ <^ is a constant and ^cis an 
 infinitesimal. 
 
 Hence by [2] Zl ^ is that constant which differs from 
 the variable z!l a; by an infinitesimal ^ c. 
 
 Therefore by the definition Zl ^ is the limit oi ^ a. 
 
 Again ^ 
 
 [3] PH — P'H' = PK. 
 
 Now again let P move along the curve till it reaches P'. 
 Then PH is a variable, P'H' is a constant, and PK is an 
 infinitesimal. 
 
 Hence by [3] P'H' is that constant which differs from 
 the variable PH by the infinitesimal PK. 
 
 Therefore by the definition P'H' is the limit of PH. 
 
 It is obvious that both the variable ^ a and the line 
 PH reach their limits. 
 
INFINITESIMALS AND LIMITS 1 5 
 
 In Fig. 4, 
 
 [4] Z. P"KD — P"EP -= Z PED. 
 
 Let P move continually to the right. Then Z- P"KP 
 is a variable, Z. P"ED is a constant and Z PED is an in- 
 finitesimal. 
 
 Hence by \_\\^ Z P"ED is that constant which difiers 
 from the variable Z P"EP by the infinitesimal Z PED. 
 
 Therefore by the definition Z P"ED is the limit of 
 Z P"EP. 
 
 Let Sn = the sum of « terms of the series i + >^ H- /4^ etc. 
 
 Let R„ = 2 — j„. 
 
 Let n increase indefinitely. Then 5 is a variable, 2 is 
 a constant, and R« is an infinitesimal. by § 5. 
 
 Hence 2 is that constant which differs from the vari- 
 able s,, by the infinitesimal R„. 
 
 Therefore by the definition 2 is the limit of s„. 
 
 It is obvious that neither of these last two variables 
 can ever reach its limit. 
 
 16. Hence there are two classes of Limits : 
 
 [i] Those to which the variable finally becomes equal. 
 [2] Those to which the variabl e can never become equal . 
 The first class are called Attainable Limits. 
 The second class are called Unattainable Limits. 
 
 17. In Fig 4, 
 
 let V = the variable Z P"EP, 
 let / ^ its limit Z P"ED, 
 and e = the infinitesimal Z PED. 
 [i] Since Z P"EP = Z P"ED — Z PED. by Geom. 
 [2] V ^= h — c 
 
 And in general whatever the variable may be, 
 
1 6 INFINITESIMALS AND LIMITS 
 
 [3] 2; = / — £. 
 
 In [3] € may be either positive or negative. For if as 
 in Fig. 4 the variable always increases as it approaches 
 its limit € will be negative, but if in any case the vari- 
 able always decreases as it approaches its limit c will be 
 positive. There are some cases in which the variable 
 alternately passes from one side of its limit to the other 
 as it approaches it. In these cases e is alternately posi- 
 tive and negative. 
 
 PROPOSITION I 
 
 18. If two variables are always equal, and each 
 approaches a limits their limits are equal. 
 
 Let V and v^ be two variables, 
 
 / " /' be their respective limits, 
 
 and € *' c' be two infinitesimals. 
 
 Let V =^ v' 
 
 We are to prove that I =^ I'. 
 
 [i] V = I — € by § 17. 
 
 [2] and z/'=/' — e'. by § 17. 
 
 Subtracting [2] from [i] we get 
 
 [3] V — v' = l~l' —€^ €! 
 
 [4] But by hypothesis v = v\ and hence v — v^ ^= o. 
 
 Substituting this value oi v — v' into [3] 
 
 [5] we get o = / — /' — € -{- c'. 
 
 [6] Hence /' — /=€'—£. 
 
 If € and e' are not equal to each other suppose that 
 the absolute value of c > the absolute value of e'. 
 
 [7] Then c' — e is not greater than 2€, 
 
 and from [6] it follows that 
 
I 
 
 INFINITESIMALS AND LIMITS 1 7 
 
 [8] /' — /is not greater than 2c. 
 
 Since by § 15 / and V are constants, /' — / is a constant. 
 And, since by § 5 e can be made as nearly zero as we 
 please, 2c can also be made as nearly zero as we please. 
 Then it follows from [8] that V — / is a constant which 
 can never be greater than 2c. But the only such constant 
 is zero. 
 
 [9] Hence V ~ I = o. 
 
 [10] or /' — /. Q. K. D. 
 
 PROPOSITION 2 
 
 19. If the sum of any finite number of variables be 
 variable, then the limit of their sum is equal to the sum of 
 their limits. 
 
 Let V and v^ = two variables. 
 
 / " /' = their respective limits. 
 
 and e " c' = two infinitesimals. 
 
 We are to prove that lim (z;4-2;' )=/-(-/'. 
 
 [i] 7; = /— c by § 17, [3] 
 
 [2] and v'=l'-^ by § 17, [3] 
 
 [3] By addition 2; -f- z^' = / -f /' — e — c' 
 
 [4] and limit (z; -f z;') = limit (/-{-/' — c — c'). 
 
 by § 18. 
 
 Since by hypothesis e and e' are infinitesimals, 
 ( — c — c') is an infinitesimal by § 12. 
 
 Also since by § 15 / and /' are both constants, / + /' is 
 a constant. 
 
 Hence / + /' is a constant which differs from the vari- 
 able [/' 4- / — e — e'] by the infinitesimal [ — c = c']. 
 [5] Therefore limit (/-(-/' — £—€') = /-[- /' 
 
 by § 15. 
 Substituting into [4] we get 
 
1 8 INFINITESIMALS AND LIMITS 
 
 [6] limit {v -\- v') = I ^ I'. 
 
 Q. E. D. 
 
 Similarly the theorem may be proved for the sum of 
 any finite number of variables since (e — e' — e" . . . etc.) 
 is an infinitesimal. by § 12. 
 
 PROPOSITION 3 
 
 20. If the product of a finite number of variables be 
 variable, then the limit of their product is the product of 
 their limits. 
 
 Let V and v' = two variables, 
 
 / " /' = their limits, 
 
 and € *' e' = two infinitesimals. 
 
 We are to prove that lim vv^ = //'. 
 
 [i] v= I — € by § 17, [3] 
 
 [2] v'=l'=€ '' *' 
 
 Multiplying [i] by [2] we get 
 
 [3] z;z/ = //' — €/' — €7 4- ec' 
 
 [4] Therefore 
 
 limit {vv') = limit (//' — c/' — eV + «V). 
 
 by § 18. 
 
 Now €/' and c7 are infinitesimals by § 11. 
 
 and ee' is an infinitesimal. by § 13. 
 
 Hence — d' — eV -f ee' is an infinitesimal, by § 12. 
 
 But since by § 15 / and /' are constants //' is a constant. 
 
 Now //' is a constant which differs from the variable 
 ir — e^' — eV -{- ee' by the infinitesimal — d' — e7 + ee'. 
 
 [5] Hence lim (//' + «/' — ^7 -f- ee') = //' by § 15. 
 
 [6] Hence by [4] lim {vv') =^ II' q. E. d. 
 
 Similarly the proposition may be proved for the product 
 of any finite number of variables. 
 
INFINITESIMALS AND LIMITS 1 9 
 
 21. Corollary i. — The limit of the product of a constant 
 and a variable is the product of the constant and the limit 
 of the variable. 
 
 Let a = any constant 
 " z/ = any variable 
 and / ^ its limit 
 Let e = an infinitesimal, 
 [i] Then v = l—€ by §17, [3] 
 
 [2] av= al — a€ 
 
 [3] Hence lim (av) = lim {al — ae). by § 18. 
 Now a€ is an infinitesimal by § 11. 
 
 but since by § 15 / is a constant al is a constant. 
 
 Hence al is a constant which differs from the variable. 
 (al — ae) by the infinitesimal ac. 
 Therefore lim (al — ae) = al by § 15. 
 
 Substituting into (3) we get 
 lim {av) = al. 
 
 Q. E. D. 
 
 22. Corollary 2. — If the product of any finite number of 
 variables be a constant the limit of their product is the same 
 constant. 
 
 Let v,w,x, . . . etc. be variables and let « be a constant, 
 [i] Let vwx . . . etc. = a. 
 
 We are to prove that 
 
 limit {ywx . . . etc.) = a 
 
 Let 2 be another variable 
 [2] Then by (i) (vwx . . . etc.) z ^= az 
 
 [3] lim (vwx . . . etc.) lim z =^ a lim js 
 
 by § § 18 and 21. 
 [4] .*. lim (vwx . . . etc.) = a 
 
 Q. E. D. 
 
20 INFINITESIMALS AND LIMITS 
 
 PROPOSITION 4 
 
 23. The limit of any positive integral power of a vari- 
 able is the same power of the limit of the variable. 
 
 Let V = the variable, 
 
 / = its limit, 
 and ;? = any positive integral exponent. 
 We are to prove that lim v^ = V 
 
 [i] limit {yvvv ... \.o n factors) = /././. ... \.o n fac- 
 tors, by § 30. 
 [2] Therefore limit if' = I". 
 
 Q. E. D. 
 
 PROPOSITION 5 
 
 24. The limit of the quotient of two variables is the quo- 
 tient of their limits^ provided that neither of the limits be o. 
 
 Let X and y be the two variables. 
 
 we are to pre 
 
 >ve tnai nm — = r^ 
 
 y \\my 
 
 
 [I] 
 
 Letz/= -^ 
 
 y 
 
 
 [2] then 
 
 vy =^ X 
 
 
 [3] and 
 
 lim {vy^) = lim x. 
 
 by § 18. 
 
 [4] But 
 
 lim {vy) =lim vVimy 
 
 by § 20. 
 
 [5] hence 
 
 lim V lim y = lim x 
 
 
 [6] and 
 
 ,. lim;t 
 
 lim V = 7: 
 
 lim y 
 
 
 [7] or by [I] 
 
 ,, X lim x 
 
 lim — = r- • 
 
 1/ lim 1/ 
 
 
 Q. E. D. 
 
INFINITESIMALS AND LIMITS 21 
 
 25. Corollary, — The limit of the quotient of a constant 
 by a variable is the constant divided by the limit of the 
 variable. 
 
 Let a ^ any constant, 
 «< 2, ^ *< variable, 
 and / = its limit. 
 We are to prove that 
 
 [i] Let 
 
 [2] Then 
 
 [3] Hence lim zv = a by § 22. 
 
 [4] and lim 2 lim v ^= a by § 20. 
 
 [5] hence 
 
 [6] or by [i] lim -^ == ^r:^ — 
 
 V 
 
 lim V 
 
 2 = 
 
 a 
 
 V 
 
 2V = 
 
 a 
 
 lim 2v = 
 
 a 
 
 im 2 lim v = 
 
 a 
 
 lim 2 = 
 
 a 
 
 \\mv 
 
 I1 fr\ — 
 
 a 
 
 Q. E. D. 
 
 PROPOSITION 6 
 
 26. The limit of the nth root of a variable ts equal to the 
 nth root of the limit of that variable. 
 
 Let V = any variable 
 
 and / = its limit. 
 
 n n 
 
 [i] z/ = y^z;" and /= y^'/", 
 
 [2] also limit v = I by hyp. 
 
 Substituting in this equation the values of / and v 
 found in [i] we get 
 
 [4] limit \/lF ==. ylF 
 
22 INFINITESIMALS AND LIMITS 
 
 Now since v represents any variable whatever, v'' rep- 
 resents any variable whatever and /" is its limit. 
 
 by § 23. 
 Therefore [4] shows that the limit of the «th root of 
 any variable is equal to the nih root of its limit. 
 
 Q. E. D. 
 
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