144 



AI.GEBRA. 



and now xvz may be considered a single variable and a be its 

 limit, and a repetition of the application of the theorem of the last 

 article would show that the limit of the product oi four variables 

 equals the product of their limits, and evidently this reasoning- 

 could be carried as far as we wish. 



12. ThKOKEM. The limit of the quotient of two variables equals 

 the quotient of theif limits. 



With the same notation as before, we are to prove that 



.. X a 

 lim -=7-. 

 y b 



^ X . 



Let — =^, then x=^qy. 



y 



.". lim -r=lim (qy ) = lim. ^.lim r; 



lim X a 



.'. lim^=^-^ =-. 



lim y b 



13. Theorem. The limit of the reciprocal of a variable equals 

 the reciprocal of its limit. 



With the same notation as before, we are to prove that 



lim (il = i. 



U") a 



We know that ~x-^=^x, 



X 



hence lim — jr* = lim x=a. 



But lim -x^ = lim - 1 . lim x" 



X ] \x 



= «^ lim 



x\ 



hence «^ lim ~ = 



Lrj 



hence lim 



^ x\ a 



14. Theorem. The li?nit of a7iy power of a variable equals that 

 poiver of the limit of the variable. 



With the same notation as before, we are to'prove 



lim jr"=a", 

 n being any commensurable number either positive or negative, 

 integral or fractional. 



