128 INDETERMINATE FORMS. 



153. For example, required the value of 

 1 



when as = 0. 

 cot x 



Differentiating both numerator and denominator, we find 

 the required limit is the same as that of 



_1_ 



a? . sin 2 a; , . . 



or ot , that is, unity. 



sn' x 



The same result may be obtained by writing the proposed 

 fraction in the form - ; thus 



1 



x tana; 1 sin a; 



^^^^^_ __ _ f\t* ~ ^^~~ 

 - ui - 



cot a; x cos a; x 



sin 05 



The, limit of - is 1. and the limit of - is 1 : therefore the 

 oos a; a? 



limit of the proposed fraction is 1. 



a?" 

 As another example we may find the limit of -= when x is 



infinite, n being positive. 



/p** 7it w 



The limit of ^ = the limit of 



*u r -4. f n ( n ~ !) ajn ~ a 

 = the limit of - - ~ - . . . 



e 



Proceeding thus, we shall, if n be a positive integer, arrive at 



[n 

 the fraction ^ , the limit of which is 0. If n be a fraction, 



6 



we shall arrive at a fraction having e x in the denominator and 

 some negative power of x in the numerator, which also has 

 for its limit. 



x* 

 Hence the limit of , when x = oo , is zero. 



6 



