124 INDETERMINATE FORMS. 



These results may also be obtained without the use of 

 Taylor's Theorem. 



If < (a) = and ty (a) = 0, we have 



<ft (a + h} $ (a) 

 <f> (a + ft) _ <ft (a + K) - <j> (a) _ h 



ty(a)~ ty (a + h) - $ (a) ' 



h 

 Now diminish h indefinitely, and we have 



,, i. ... c ^fa) 0'( a ) 



the limit when x = a of - 7-7 is T, ; . . 

 TJT (x) ^ (a) 



If </>' (a) = and ty' (a) = 0, we have in the same way 



,<'(#) . <j>"(a) 



the limit when x = a of ,,. ' is ;; : . 

 ^(x) +'($ 



Hence, the limit when x = a of -?4{ is -?jA-T 



^(*) ^r (a) 



This process may be extended, giving the same result as 

 in Art. 146. 



148. Form . 



00 



Let <f>(x) and ty(x) be functions which both become infinite 

 when x = a ; it is required to find the limit of the fraction 

 *(*) 



f (*) ' 



i 



and 

 when # = a; hence, by the previous rules its limit is 



and the fraction on the right-hand side takes the form - 



Hence 



therefore 



- 

 () 



