Theory of Limits. 143 



.9 Theorem. TJie limit of a constant multiple of a variable 

 equals that co7istant multiplied by the limit of the variable. 



Let X be a variable and a its limit. We are to prove 

 lim nx—7ia. 



Let a—x=^u. Then x may be taken so near to a as to make 



k 

 n 

 or nu<ik, 



however small k may be. 



a — x=u .'. na — nx==7iu, 

 hence 7ix may be made to differ from 7ia by an amount as small 

 as we please. Yet 7ix can never equal 7ia\ el.se x could equal a. 

 Hence lim 7ix=na. 



10. Theorem, ^lie liifiit of the p7'oduet of tivo variables equals 

 the product of their li77iits. 



With the same notation as in Art 7 we are to prove that 



lim xy=ab. 

 xy=(a — u)(b — v)=ab — av — bu-j-uv 

 =ab— ( av -\- bu—uv ) . 

 Since lim v=o, lim az'=o, 



and as lim u=o, lim b7i=o, 



and since u and v are each as small as we please and the product 

 smaller than either, lim uv==o. And since the limit of each term 

 oi av+bu — uv is zero, the limit of the algebraic sum of all three 

 terms is zero. Hence xy may be made to differ from ab b)^ an 

 amount as small as we please ; hence lim xv=ab. 



11. Theorem. 77ie li7nit of the product of a7iy nu77iber of va7'- 

 iables is equal to the product of their limits. 



With the same notation as before we are to prove 



lim (xyz . . . )=^abc . . . 

 We have already proved that 



lim xy=ab, 

 and we may consider xy a single variable and ab its limit ; then 

 by the last article 



lim (xy.zj==.ab.c, 

 or lim xyz=abc, 



