DIFFERENTIAL AND INTEGRAL CALCULUS. 73 



which is of the form - , and to be evaluated the same way. 

 If more convenient, we may reduce to the form , which 

 has been shewn to be subject to the same rule. 



(3) The form co - co ; if f(a] = co and <f> (a) = co , then 



now if the limit of ^ { be 1, this is of the form co x 0, 



/() 

 and may, therefore, be treated as in (2), and if the limit 



of yr- r be different from 1, the limit of f(x)- < (x) is 

 co . Thus 

 seca?- i&ux = secx 



and when x = ATT. the limit of this is [ 



\- 



So also -= -cot 2 x = , [1- -- =-)- which since we 

 x* x* \ tan 2 ic/ ' 



know that ( - ) is 1, takes the form co x 0, and we 



Vtana?/^ 



may write this 



/ x V 2 

 ' Vtana?/ ' 



sin x - x cosx amx +x CQSX 



x* tan 2 ic x* x 



the product of three fractions wMch all take the form - 



when # = 0, but of which we know the limit of two ; i. e. 



( x \ , f&mx-\-xcosx\ /sina? \ 



- =1; and - - = [- + cos# =2. 



Vtan^;^ V x J X=Q \ x ) x ^ 



The remaining one 



/since x cosafv _ /cos x cosx -f x sina? 



v v- 



(1 \ 2 



COt a J3j = - . 



The three indeterminate forms 1 s0 , co, can be reduced 

 to forms already discussed, by taking the logarithm in 



L 



