DIFFERENTIAL AND INTEGRAL CALCULUS. 77 



( p.T H gWUB 



) is co , or n GO when x. Now 

 X / X 



1 e mx 



if we put s x = - , then when x = GO , 2 = 0, and becomes 



since the limit 





_ 1) ., (logg 



of this when x co , and, therefore, when z = Q is co , 

 therefore that of z m (logz) n is 0, or the limit of x m (\ogx) n 

 when x = is 0. 



In conclusion, it may be mentioned that any one who is 

 familiar with the ordinary expansions in series of Algebra 

 and Trigonometry, will find it in most cases easier to 

 determine all such limits without the aid of the Differential 



Calculus at all. They should be reduced to the form - 7 



which can always be done, as has been seen, and if this 

 happen when x = a, put a + z for #, and expand numerator 

 and denominator in powers of z. The only failing case 



is of such functions as logcc, ~ , e"^... when a? = 0, but 

 these are quite as troublesome, if a strict proof is required, 

 when the Calculus is used. 



The following is an example given in Todhunter: 



(0 + sin 6 -4 sini0) 4 



rais > which would require 12 dif- 



(3 + cos0-4cosi0) 3 (e=o1 ' 



ferentiations if solved by the rule ; now 

 + sin0-4 sini0 



/33 ,Q A3 v m 



= 6 + 0- -+...- 4 (^ -- -K..J = -y- 2 + higher powers,. 

 3-i-cos0-4 cos0 



= 32 + ^ er P owers ? 



. . (32) 3 2 15 2 7 128 



hence the required limit is ^ = 34-^8 = ^ or 



