DIFFERENTIAL AND INTEGRAL CALCULUS. 75 



Hence, the only case in which functions which take the 

 form or co can have any other limit than 1, is where 

 one or both of the functions vanish when x a^ but do 

 not admit of any expansion in positive powers of x a. 

 Such forms are 



-i -i 

 (x a) r log (a; a), s*"", ec*-^ 2 , &c. 



The forms 0, GO , may then usually be interpreted 1, 

 although one may with some trouble invent cases in which 

 their limits differ from 1. Thus, if 



u = s x and v = 1 



and the limit of this when x = is s~' 2 . 



In the case I 00 , if when x = a^ y (#) = !, <(#) = GO ? 



and y = {/(x}}* X) , logy = g ^^ , and its limit is that of 



/w 



I/ST -_. W 

 ' 



limit of 



-_ 

 *(*)'/ /' 



Now (x a)f(x) takes the form x GO when x a, 

 and its limit = limit of ^^ = limit of - J /, ' J ; there- 



7W 



fore, if m be this limit, that of log?/ is m<$ (a), or that 

 of y is =""^ fl) . Such forms can also generally be made 



to depend on the limit of (1 + 0)r ? when 2 = 0, which has 

 been investigated in the first chapter, and whose limit is 

 m . For example, 



(cos*)* 2 = (cos'^K = (1 - sin^>* 2 = {(1 - wfx)* *} *? , 

 and the limit when a? = is therefore ~. Similarly the 



m m 



.. . f /smic\^ , /tantcX^ , . , f 



limits of - - ] , and - may be found, for 

 V x J ; \ x J 



s'mx x* x* x* . 





