DIFFERENTIAL AND INTEGRAL CALCULUS. 



or 



1 + H --- + 



2 



2.3 



+ 



2.3.4 



-f- . . , 



But the limits of 1 , 122, 1 -30..., 1 nz, are 

 all 1 when z = 0, for all finite values of n ; hence the 

 required limit is the limit of the sum of the series 



a certain abstract number which is usually denoted by g. 



HenCG Tx ( l 8* x ) = x lo ' or xlotra ' the base 

 supposed when none is written. 



Now to find ( j ) , take the numerator =M, and 



> ll '** 

 therefore 7i = log a (l + M), and when 7i = 0, zj = 0, hence 



fa h -l\ ( u 1 



7 = -Ji - - - -\ , from the limit lust 



V h /, =0 llog^tl^-u))^ log fl 



previously found; whence -- (a] =-. - = a log a. 



Dfl 



y = sin mx ; A?/ = sin m (x -\- h) sin mx 



. mil 



_ . mh f mh\ 



= 2 sin cos f mx + 1 ; 



Ay 



or 



cos 



mh 



-jt = m cosmx. the limit of 7 beinsr 1, 

 ax mn 







(5) y = coswzir, 



p 



Ay = cos w (a? + A) - cos???ir 7 = 2 sin sin f mx + - - - ) , 



