INDETERMINATE FORMS. 135 



Thus e v takes the form - when x = a, and its value may be 



investigated by Art. 145. 

 Or we may proceed thus, 



ylr (x) 



then y is infinite unless the limit of ^ . is unity ; if the 



limit of 7-r-r is unity, 

 <f> (x) J> 



since 



\s 



*(*) 

 it takes the form - . 



For example, suppose y = tan x sec x ; 



7T 



then ti takes the form oo oo when a; = . 



2 



A i /'t sec ^ 



Also y = tan # 1 



V tan x) 



_ 1 cosec x _ 

 cot a; ' 



this takes the form - , and its limiting value is 



cosec x cot x - 

 - 2 or 0. 

 cosec x 



161. The limit of " when a; = co, supposing F(x] to 

 x 



F'(x) 

 be then infinite, will be the same as that of p-^, or F'(x). 



See Art. 151. 

 But, 



If x be made to increase indefinitely the limit of the 

 second member of the equation is "(x). 



