INDETERMINATE FORMS. 123 



146. Suppose that not only 



<f> (a) = 0, and Y (a) - 0, 



but also <=0, <'=0, ...</>" (a) = 0, 

 and -^' (a) = 0, i/r" (a) = 0, . . .ty n (a) = 0. 



By Art. 92, 



- < (a) - Af (a) ... - ^W- 



Hence, by division, we have 



<f) n+1 (a+0h} 



Dimmish h indefinitely, and we have 



, <j> (x) . </>" +1 (a) 



the limit when x = a 01 , , ( is , n+1 , > . 

 t(*) t l) 



147. In Art. 145, if 



t=, 

 and </>' (a) = some finite quantity, 



<f> (a;) 



we have the limit when x = a of ; . ( is infinity ; 



^(x) 



if < = 0, 



and i|r' (a) = some finite quantity, 



6 (x) 



we have the limit when x = a of ; is zero. 



Y(*) 



And in the same manner, we may shew that if the first 

 of the differential coefficients <' (a), <f>"(a), ... which does not 

 vanish, is of a lower order than the first which does not vanish 



of the series i|r' (a), -fy" (a), ..., the limit of y--W when x = a, 



Y ( x ) 

 is infinity ; if of a higher order the limit is zero. 



