136 INDETERMINATE FORMS. 



F(x] 



Hence the limit when x = oo of " 



x 



,, r .. , e 



= the limit when x = oo of 



. 

 h 



If for simplicity we make h = l, we have 



the limit of ^-& = the limit of [F(x + 1) - F(x}}. 



ffs 



162. The limit of {F(x}}" when x is infinite, is the same 



log Fix) 



as that of e * . 



But, by Art. 161, supposing F(x) to become infinite with x, 



the limit of - - is the same as the limit of 

 x 



of 



_ 



Hence the limit when x = x of {F (x)}* 



., r .. f F(x + l) 

 = the limit of , \ 

 F(x) 



Suppose, for example, that we require the limit when x is 



of)* 



infinite of }. 



IL^J 



By the theorem just proved the required limit 

 = the limit of 



\x + 



/, 

 = the limit of 



x 



= the limit of ( 1 + - ) 

 V xj 



e by Art. 16. 



163. A few remarks may be made on indeterminate frac- 

 tions involving more than one variable. 



