10 EXAMPLES OF LIMITS. 



18. Since the limit of ( 1 + -I when x increases indefi- 



/ I 

 ( 1 + -I 

 V xj 



I 



nitely is e, we see, by putting - = z, that the limit of (1 + z)* 



sc 



when z is diminished indefinitely is also e. Hence we can 



i 



deduce the limit when z = of (1 + az)*, where a is any 

 constant quantity. For 



Now as z diminishes without limit, so also does az, therefore 



j_ 

 the limit of (1 + az] at is e, 



i 

 and the limit of (1 + az}* is e a . 



19. Since log a (1 + *) *= - log a (1 + z\ 



Z 



a being any base, we have, by diminishing z indefinitely, 



the limit of Io 8( 1 + *) = the limit of log a (l + jr)', 

 z 



= loge; 

 and, putting e for o, 



the I limit of 



z 

 20. From the equation 



we deduce, by assuming 1 + z = a", 

 lo !+ 



~a-l* 



Now suppose z to diminish without limit, and therefore also v. 

 We have then 



the limit of - when v = 

 a 1 



i 



= limit of log a (1 + z) * when 2 = 



