INDETERMINATE FORMS. 137 



A function of two variables may tako the form - , either 



when one of the variables remains undetermined and the 

 other has a particular value, or when both receive particular 

 values. 



As an example of the first case, suppose 



( \ _j_ I N 2 ' 



if we make x = a we have z = - , whatever y may be. But 



by removing the factor x a from the numerator and deno- 

 minator of z, we have 



_ c (x + a) 



~ y + x-a ' 



Hence, when x = a, we have 



. 

 V 



This case is very simple, and whenever it occurs the ap- 

 plication of the preceding rules will give the limiting value 

 towards which z approaches as x approaches its limit. 



As an example of the second case, suppose 

 _c(x-d) 



Z=~ 7 * 



y-b 



This fraction takes the form - when x = a and y = 5, and 



is really indeterminate. For suppose y ft = m (x a), then 



c 



z = . 

 m 



Hence the value of z is indeterminate, for x and y being 

 independent m may have any value we please. 



164. It may happen that the values which such a function 

 assumes, although infinite in number, are confined within 

 certain limits. For example, suppose 



_ c(x-a)(y-V) 



