( 122 ) 



CHAPTER X. 



LIMITING VALUES OF FUNCTIONS WHICH ASSUME AN 

 INDETERMINATE FORM. 



144. IN the statement, the limit of - - = 1 when Q 







diminishes indefinitely, we have an example of a fraction 

 which approaches a finite limit when the numerator and de- 

 nominator each tend to the limit zero. The object of this 

 Chapter is to find the limit of any fraction of which the 

 numerator and denominator ultimately vanish, and also the 

 limiting value of some other indeterminate forms. 



145. Form - . 



<6 (x) 

 Suppose 



\/r (#) 



such a fraction that both numerator and denominator vanish 

 when x = a ; it is required to find the limit towards which 

 the above fraction tends as x approaches the limit a. 



We have proved in Art 92 that 



< (a + K) - < (a) = h<j)' (a + 6K), 



If then <j> (a) = and i/r (a) = 0, we have, by division, 



Let h diminish indefinitely ; then 



..-u v -i i. f 4>( x } 



the limit when x = a of , . , is 



,. . 

 a ) 



