126 INDETERMINATE FORMS. 



Suppose the limit of - . ^ is really zero ; then the limit 



of y W 9> ( x ) j s rea iiy finite, namely, unity. Hence, it has 



-^(x) 

 been proved that 



,, .. . -|r () + <j> (x) , 



the limit of T , , ; when x = a 



is 



- ,,. . , 

 a;) ^ (a) 



that is 1 + the limit of ^-= 1 +-7^ ; 



^ W ^ ( a ) 



therefore the limit of iM = ^ . 



i/r (*) ^ (a) 



<f> fa;) 

 If the limit of \ , ( be really infinity, then the limit of 



Ww 



T-T-T is really zero, and therefore, as just shewn, the limit of 



^f^> (CC) (f) \3Ci 



,,, / will be zero. Hence, the limit of ,,. . will be infinity. 

 (*) t (*) 



Combining then this Article with Ail. 148, we can assert 

 that if <f> (x) and -^ (x) both become infinite when x = a, the 



(f) (l*\ <f) fwC^ 



limit of - . , ( will be the same as the limit of ; , , : . 



151. The two Articles 148 and 150 may be replaced by 

 the following mode of exhibiting the proposition. 



Suppose ^ (a) = o> , and i/r (a) = oo . 



Then -^ =0 and L- = 0: 



</>(; ^-(o) 



now = = (Art 1Q6) . 



' 



therefore <f>' (a + 0h] ^ <[> (a 



' 



(a + A) 



