INDETERMINATE FORMS. 127 



<> (x) 



If . , ( has a finite limit when x = a, the limit of the 

 *(#) 



second factor on the right-hand side of the equation is unity. 

 Hence 



the limit of = the limit of 



. 

 f{) ^ (x) 



<> (x) 



But if , , ( tends to or oo as a; approaches a, it will in 

 -^(x) 



general finish by approaching the limit in such a manner that 

 the second factor will in the first case be less than unity, 



$ (x) 



and in the second case greater. Hence. ,,, ' becomes zero 



^(oj) 



<b(x) 



or infinity at the same time that . . . does. 



ty(x) 



152. In the preceding rules for finding the limit of a 



function which takes the form - or when x = a, we have 



oo 



made no supposition as to the magnitude of a. Hence the 

 rules are often applied to the case in which a is infinite. But 

 for a direct demonstration of this case we may proceed thus. 



<b (x) 

 Suppose the limit of JT-T required, when a; = x ; it being 



T ( X J 



known that then either ^>(x) = Q and ty (x) = 0, or < (x) = oo 

 and ty (x) = oo . 



Put x=-, then 



Now as y tends to zero, we have, by preceding rules, 



$ (-} \ f (- 

 the limit of -& = the limit of y - ^ 



I -T* 1 ! 



Ti~ 



\ // 



= the limit of . = the limit of . 



