Theory of Limits. 



149 



From ( I ) lim (a^ -^a' ) = \\m. a'~^hy Art. 7. 



But lim (a'' -^a-' j=lim a"" -r-lim or^ =a" -t-«' , 

 and lim a'~^=a"~'\ 



therefore a" -^a'' =a"~''. 



21. It is also easy to see that where // is incommensurable 



a"b"=(ab)" 



and 



Hence 

 But 

 and 

 hence 



Again 



by Art. 7. 



For let ^ and b be tvvo bases and x a variable which remains 

 commensurable, but approaches an incommensurable limit ft, then 



a-b-^=(ab)' . 

 lim ^"/5"=lim (aby 

 lim «'' ^' =lim «" lim b'- =a" b" , 

 lim (ab)-' =(ab)" , 

 a"b^ = (adj". 



b^ Ui 



hence 



and 



hence 



a 

 b" 



a 



b" 



22. Examples of Limits. In the following examples an 



limit (a—h 



expression like , v r Ms to be read: the limit ^/*i — 7-7!- 



as h approaches a as a limit. The symbol ^ stands for the 



word approaches. 



_,. . limit ( 



/. Find ^ \ 



h ^ o (. 



Process. 



limit {(x-\- /;/— A") limit f x^-\-2hx—h'—x'- 



limit f (x-\-hf—x'\ 

 ~h / 



// ^ ol 



I hmit f 

 \~h : o I 



f 



Hmit f 

 >^ : ol 



2X^h 



}="■ 



