DIFFEEENTIAL AND INTEGRAL CALCULUS. 71 



Hence prove that 



rt sin20cos' 2 sin30cos 3 

 ITT- = sin cos0-f - - -1- - ^ - + ---- 



!i o 



9. If /' (x) continually increase as x increases, or con- 

 tinually diminish, prove that f(n+ 1) -f(n) lies between 

 f(n) and/' (w + 1). 



Hence prove that 



+ 1 ) and <l + logw, 



tan^0 , tari(?z-fl)0 

 (2) sec 2 + sec 2 20 +...+ sec 2 nO > g and < -- ^j- -- 1, 



provided that (n + 1) 9 < ^ir. 



DIFFERENTIAL CALCULUS. 

 INDETERMINATE FORMS. 



If /(a?), $ (a?) be two functions of a? which both vanish 

 when# = a, the fraction f (x) -r- </> (x) becomes unmeaning, 



fix] 

 but as x approaches a, ( rr\ w iU generally tend to some 



limit from which it may be made to differ by less than any 

 assignable quantity before x a. Thus, 1 a?, and 1 x 2 

 both vanish when x = 1 ; but 



l-x I 1-x 1 1 l-x 



- - ^ = - - , and therefore - -- s - - = - . 

 \-x i 1-fa? 7 l-x' 2 21 



which as x approaches 1 diminishes, and may be made less 

 than any assignable quantity before 02 =1. Hence the 



1 x 



limit of the fraction -- s , as x approaches 1, is . This 

 J. ~~ x 



result is generally written ( - 2 ) = - , but the correct 



Vl-^/ a .=i 2 

 meaning should be carefully borne in mind. 



Now, in general, if /(a) = 0, we have, putting x = a + 0, 



/ =/(a + ) =/(a) + /' (a + 0) = */' (a + ft,), 

 and so </> (aj) = 00' (a + 0s) ; 



thcreforc 



(x) 



