130 INDETERMINATE FORMS. 



Again, suppose we have to find the limit of 



as x approaches unity ; put x = 1 + h, and the fraction becomes 

 A + 1) - l + VA 



Multiply both numerator and denominator by 



and we get 



2^/A S 



v or - 



and the limit of this, when h = 0, is -j- . 



156. There are cases in which not only <j>(x) and ty(x] 

 vanish, but all their differential coefficients, and where, con- 



<b(x] 



sequently, we are not able to ascertain the limit of T-T-V - 



+(*) 



For suppose <f>(x) =a~", where u stands for , a and n being 



C 



positive numbers, and a greater than unity: we have 

 ,. . wloga.cT" 



^ 



a;' 



^ fwloga ?? + 1) 

 a.a-" L^- I 



I *V tA/ J 



and so on. 



Put - = z, and let t stand for z n : 

 x 



,,, . ?iloga.2" +1 



then d> (a) = ^. . 



a 



,, tf \ n log a (w loer a . s 2(n+1) (n 



(D (OCl ^ 



r \ / t 



