INDETERMINATE FORMS. 129 



154. A remark should be made for the purpose of pre- 

 venting a misconception of some of the results of this Chapter. 

 Suppose <f> (x) and t/r (x) both to vanish when cc = a, and that 

 (f)'(a) = while ty' (a) is finite. We say then, that when x = a, 



the limit of ^ = the limit of ^ , 



^(x) $ (x} 



meaning that each side of the equation vanishes. It does not 

 follow necessarily that 



<(ir) <f>(x) i , f , r . 



- . . -7- ,, has unity tor its limit. 

 - 



For example, let < (a;) = cc 2 , -^ (a;) = sin x, 

 then <>' x % x > r ' x 



When x approaches the limit zero, we can infer that, since 



.,, . approaches zero, so also does ~-^, . But it is obviously 

 T(W) ' &(*) 



not true that the limit of 



a? 2x p x* cos x . 



or of - = - is unity; 

 2x sin a; J 



sm x cos a; 

 the limit is in fact . 



155. It should be observed that there are examples which 

 may be solved by means of the Differential Calculus, but 

 which can also be solved, and sometimes more simply, by 

 common algebraical transformations. For instance, 



fa-o)* 



when x = a takes the form - . Put a? =* a + k, and the fraction 



becomes 



A* 



or 



and the limit, when h = 0, is 0. 



T. D. c, K 



