72 DIFFERENTIAL AND INTEGRAL CALCULUS. 



and therefore limit of 



or, more correctly, limit^ = limit^ - 



This equation is the fundamental one for determining 

 the limits of functions which assume an unmeaning form. 

 Iff (a) and <j>'{a) do not both vanish, the limit is/'() -=- <'(), 

 but if they do, we must continue the process, and we shall 

 have 



and so on so long as fo^ numerator and denominator 

 vanish; the limit being attained when one or both are 

 finite. Thus, if /(),/' (a).../~* (a) ; (a), <' (a)...^' (a) 

 all =0, and/" (a), (f> n (a) be one or both finite, the limit of 



/M ;,/*(") 

 *(*)*' 



(1) All other unmeaning forms can be reduced to this, 

 thus, if /(a) = co and < (a) = co , 



^VA = -r T-T -:- 77-^ , which is of the form - ; 



' 



and, therefore, 



/M 



l f * 



*>L " u* (*)) 



Hence, if the limit be finite, we shall have, dividing by 

 r * WfcP r * [*'( 



limit TTT-rr limit M-Sr 



so that in this case the same rule applies as for a fraction 

 of the form - . 



(2) The form cc x ; if 



