WHEN INDETERMINATE IN FORM. 171 



The finite limiting values of - are given by 



therefore * = 0, or ^ = r . 



x x o 



And since the given equation may be put in the form 



x\ 3 (x\* i. f x \ 



-} +a - +6 - -y = 0, 



\yJ \yJ 



we see that - = ultimately satisfies it ; 



*/ 



therefore - = oo ultimately for another value. 

 x 



At ft 



Hence the limits of - are 0, or =- , or oo . 

 x b 



This method is free from the difficulty which is pointed 

 out at the end of Art. 191. 



If we wish to ascertain by the method of the present Article 

 the value of -7^ at a point for which x = a, y = b, we may put 



CL'JC 



a + x' for x and b + y' for y in the equation which connects 

 x and y. "We shall then have to find the value of -, when 



ce'=0 and y'= ; and this may be ascertained by the method 

 shewn in the preceding Examples. 



EXAMPLES. 



/ /z* w*\ 



1. If u= . /(-a *-*]) where z and y are functions of x, 



c. j du 

 find -f- . 

 ax 



2. If u = sin" 1 - , where z and y are functions of x, find -y- . 



y dx 



