VII. SPECIAL APPLICATIONS OF 

 ANALYSIS. 



7.10 Indeterminate Forms. 



7.101 -. If T assumes the indeterminate value - for x = a, the true value 



of the quotient may be found by replacing /(*) and F(x) by their developments 

 in series, if valid for x = a. 



Example : 



sin 2 x 



[si 

 i - 



sin 2 x 



i cos x ^ x 2 x* i x 2 



!Tl~^l + ' ' 2! ~ 4?* ' ' ' 

 Therefore, 



[sin 2 x "1 

 i - cos #J x =o~ 



7.102 L'Hospital's Rule. If f(a + h) and F(a + h) can be developed by Taylor's 



Theorem (6.100) then the true value of < for x = a is, 



F (XJ 



/'(a) 

 F'(a) 



provided that this has a definite value (o, finite, or infinite). If the ratio of the 

 first derivatives is still indeterminate, the true value may be found by taking 

 that of the ratio of the first one of the higher derivatives that is definite. 



7.103 The true value of - for x = a is the limit, for k = o, of 



pi 

 where / (p) (a) and F <) (a) are the first of the higher derivatives of f(x) and F(x) 



that do not vanish for x = a. The true value of J , . for x =a is o if p>q, OT if 



f (p) (a) 

 p<q, and equal to J p (p) if p = q. 



