133 INDETERMINATE FORMS 



Assume y 6 = m (x a) ; 



. cm c 



therefore z = ^ - = . 



+ 



TO 



Here the greatest value of z is when m = 1, and z always 

 lies between - and - . 



a a 



165. We give two more examples. 

 1st. Let 



(x-a)*+c(y-by 

 this takes the form - when x = a and y = I. 



Put x a*=K and y b = k; 



therefore 



If now we assume k = Ah a , we have 



and, according to the different hypotheses we make respecting 

 a, m, p, ..., we shall obtain for z finite, infinite, or zero 

 values. 



(x -y}a n -(a- y} x n +(a- ac) y* 



2nd. Let z=- ^ sV v/ H JL - 



(x-y}(a-y}(a-x) 



If x = a, and y = a, this takes the form - . Put a + h and 

 a + k for x and y respectively ; we shall have 



z = 



(h - k} kh 



If we expand (a + A)" and (a + k)", and make some 

 reductions, we obtain 



