74 DIFFERENTIAL AND INTEGRAL CALCULUS. 



each case; for if y = u v , \ogy = v log?, or = s, which 



becomes either - or . 



GO 



The two latter cases are not essentially distinct, one 

 being the reciprocal of the other ; and the limit is in nearly 

 all cases 1. For taking v =f(x), u = (f) (a?), 



log<H*) 

 log?/ =/() log</> (x) = & y , 



/w 



and its limit is therefore = the limit of ^ r-4 + , . X A or 

 log?/ = - limit of ?rp-x . "T-T-T -f( x }- Now ^ f( a ) = an( l 

 $ (a) = 0, the limit of Q-{ = limit of ^ , , ; , and therefore 



<p (X) <p \X) 



$ (&} fix] 

 the limit of *, ' J . ^j~4 is 1 in all cases in which the limit 



of '^~~ is finite ; and therefore in all such cases the limit 

 of log?/ is 0, or that of?/ is 1. In fact, it appears that the 

 limit of y is 1 in all cases in which the limit of -?T-|-{ . ^ s 



is finite; now since both/(ce) and <f>(x] vanish when x a, 

 they can both, generally, be expanded in positive powers 

 of x , and in such case if ?, n be the lowest powers of 

 x a in /(#), (/> (x) respectively, we shall have 



f(x) =A(x- a) m (] + 0), (x) = B (x - )" (1 + &}, 

 where 0, & vanish when x = a ; therefore 

 /' (x)=mA (x-a) m - 1 (1 + /.), <' (x) = nB(x- a)"' 1 (1 +/,'), 

 where /a and /// vanish when a; = a ; therefore 



*(*}/'(*) (! + ') (!+/*)' 



and the limit when x = a is , so that the limit of ?/ is 1 . 



m ' J 



