DIFFERENTIAL AND INTEGRAL CALCULUS. 7 



since these limits each = 1 when h = 0, we have limit 

 a h -\\ , dy 



and =a 



(3) y = log a x, y&y = log a (x + A), 



\og a x = log a 1 4 - , 



Ay 1 , / h\ 1 (h I h 1 h } 1 



== 7 1 I 1 H = 7) ~ ~2 ~1~ n ~~3 ?' -j 



Ao? A V JB7 Ala; 2 ^ 3 x LOO: a 



x \ / O 



] f 1 A 1 A" 



x loga ( 2 a; 3 a? a 



whence -J- = : , the series in brackets being easily 



ax x log a 



proved as in the last to reduce to 1, when the limit is 

 taken. 



(2) and (3) can be obtained independently for such 

 students as have not read the exponential series. 



-7- (a*) has been shewn = a x f , J , 

 d , 1 , 



We will find the latter limit first, as the former is 

 immediately deducible from it. 

 Putting h xz, we have 



-k&U-H or ^log (! + )', 



t//V ft/ 



and since when A = 0, = 0, we want to find the value 



of the limit (1 + z)^. 



Now, since z is to be ultimately 0, we may assume it 



i 



<1, and therefore expand (1 4- zf by the Binomial Theorem, 

 giving us 



1.2 



