INDETERMINATE FORMS. 131 



also the value x = corresponds to z = cc . But it is easy to 

 see that every expression of the form 



a 1 ' 



where a, m, n, are positive numbers, and a greater than unity, 

 is zero when z is infinite. For if we apply to this example 

 the rule for finding the value of a fraction which assumes the 



form - 00 - and differentiate r times successively, r being the 

 integer next above m, we have 



z m k 



the limit of -7 = the limit of . . . . 

 as ty (z) 



where k is some constant, and i/r (z) a function of z which is 

 infinite when z is infinite. Consequently, all the differential 

 coefficients of < (x) vanish when x = 0. 

 If then we have 



where v stands for v , and 6 is a positive number greater 



C 



than unity, and v also positive, the differential coefficients of 



<f> (x) 

 all orders of the two terms of the fraction . . ; will vanish 



when x = 0, and the limit cannot be found by this method. 

 In the case of ,v n, the fraction becomes 



this, when x = 0, will be or oo , according as a is greater or 

 less than b. 



157. The fraction 



x 



takes the form - when x = 0. Put x = - and we have 4 > the 

 y e v 



limit of which, when y is infinite, is 0, by Art. 153 ; 



K2 



