14 INVESTIGATION OF A LIMIT. 



In (1) put $=--, and raise both sides to the power y; 



then 



(1 V? / IV 



1 + - is greater than I 1 + - ; 

 A-7/ V 7/ 



tliat is, if 8 be greater than 7, 



(1\ 8 / IV 

 1 +-K is greater than I 1 + - L (3). 

 Oj \ 7/ 



/ IV 

 From (3) we see that ( 1+ - J continually increases as x 



increases. It does not, however, pass beyond a certain finite 

 limit; for in (2) write for y9, and raise both sides to the 

 power 7 ; then 



( H j is less than ^ if 7 be greater than I. 



Hence, if we put 7=2, we find that [1 + -) can never 



\ xl 



exceed 4. By ascribing to 7 greater values we shall obtain 



/ IV 

 a closer limit for 1 + - . If we put 7 = G we see that 



\ xj 



f IV /fi\ c 



(1 + -) must be less than (-) , and therefore less than 3. 

 V &/ Vv 



Since then the limit of [ 1 + - j , as x becomes indefinitely 



\ 3C/ 



great, must lie between |1 + -J and I ), where n has 



V n) \n IJ 



any positive value, we may, by ascribing successive integral 

 values to n, easily approximate to the numerical value of the 

 limit. 



