76 DIFFERENTIAL AND INTEGRAL CALCULUS. 



where u vanishes with se, and 

 tana? 





f1 



(! + '),. 



where u' vanishes with x ; therefore 



and the limit is accordingly s~^, that of (- ] being si. 



\ x / 



The two following limits, each of which may be made 



B" X 



to depend on the other, are important : x m (logo:)", 5- ; 



x 



the first when a? = 0, the second when # = co, r/z, n being 

 both positive. Now 



and limit 



=0. 



/logaft 



Q , 



So also -^r = ( SZ 

 " 



until the index in the denominator becomes either or 

 negative, when the limit appears and is cc . 



These proofs are, however, unsound, as since the result 

 in neither case is finite, the proof of the rule for finding 



fails. They are best proved by ordinary algebra, thus 



m 



Now 



and, therefore, when x is indefinitely increased, the sum. 



/ pa A 



of this series is indefinitely increased, or ( ) is cc 



\ ^ /^ 



