INDETERMINATE FORMS. 125 



149. From the last Article it would appear that the limit 

 of a fraction which tends to the form , may be found by 



considering the ratio of the differential coefficient of the 

 numerator to the differential coefficient of the denominator. 

 But, by Art. 113, when for a finite value of the variable a 

 function becomes infinite, so does its differential coefficient. 

 Hence, if 



takes the form , 



y (a) co 



-,., , { takes the same form, 



T () 



and thus the result of Art. 148 would appear to be of no 

 practical value. It may, however, happen that the limit of 



the fraction .,. ' is more easy to settle than that of -r4-4 . 



ty (X) A/r (x) 



logx 



X 



when x = 0, takes the form . 



CO 



1 



<f> (x) x 

 Here Tn~{ = r = - x, 



the limit of which is 0. 



Hence, the limit of 2 , when x = 0, is 0. 



150. The demonstration in Art. 148, which is that usually 



6 (x) 

 given, is satisfactory only in the case in which -f-j-r really 



has a finite limit. For we divided both sides of an equation 

 by this limit which tacitly assumes that the limit is not zero 

 or infinite. 



But the demonstration may be completed thus : 



