INVESTIGATION OF A LIMIT. 9 



1 H crreater than 1 + - and less than 1 H -- , where n is put 

 x n m 



for m + 1. 



/ IV / l\ x f l\ x 



Then 1 + - lies between 1 + - ) and 1 + . 

 \ xj \ nj \ mj 



Suppose x=m+a=n /3, so that a and /? are proper frac- 

 tions, then 



/ IV f l x ~0 / 1 \m+a 



( 1 + - lies between ( 1 + - ) and ( 1 -i . 

 V ay V / \ / 



that is, between 



11+ 



{(i+iypaBdjf.+m 



IV n) J IV W J 



If x be now supposed to increase without limit, so also do 



f IV f \\ m . 



m and n. The limit of 1 + - and of 1 1 H is e, and as 



V n] \ W/ 



1 and H have unity for their limit it follows that the 



n m 



(IV 

 1 H is e. 

 x) 



17. We may shew that the limit of ( 1 H ) is also e 



V x/ 

 when x is negative and increases without limit. For put 



x = z, then we have to find the limit of ( 1 j when z 



increases without limit. 



I. 1\~* fz-V 

 Now 



z I \z-\ 

 1 + 



/l + i/ 



- J ) , where y = z- 1, 

 V y I 



Let now x increase numerically without limit, then z, and 



/ IN" 

 consequently y, do the same. The limit of [ 1 + -1 is e, and 



1 . /' IV* . 



that of 1 + - is unity, and therefore the limit of ( 1 -- is e. 



y J V */ 



