86 DIFFERENTIAL AND INTEGRAL CALCULUS. 



u a minimum, and Q TT gives u a maximum, as is obvious 

 geometrically. In case of any failure by such a choice 

 of independent variable, it is sufficient to look for the 

 maximum and minimum values which the variable selected 



can have, for if u = F(z) and z = <p (#), -=- = F' (z) $' (#), 



du 





and -7- will generally change sign when <jj (x) does, that 

 ctx 



is when $ (x) or z is a maximum or minimum, the only 

 exception being when F' (z) changes sign at the same time 

 as (/>' (x). 



DIFFERENTIAL CALCULUS (4). 



1. If, when x a^ f(x) and < (x) each =0, prove that 

 the limit of the fraction f (x) -T-</> (x), when x approaches 

 the value a, is equal to the limit of f (x) -~ <' (x). Also 

 prove that the same rule holds if f(a) = co and </> (a) = cc . 



Prove that the limits of -logf- - j , and of its first 

 derived function when x = are ^, ^ respectively. 



2. Shew how to reduce fractions which assume the 

 form co x or co co , for a certain value of the indepen- 

 dent variable, under the rule in (1). Prove that the limit, 



when x = 0, of x m (logic)" is 0, and that of $ is f . 



TV J- Al. !* * S ' miC 1 



Find the limit of , ^ when ic= ?,TT. 



(JTT sc) cosic cos a? 



3. Prove that the evaluation of ihe limits of functions 

 which take the unmeaning forms of 1*, co, can be 



made to depend on functions which take the form - , and 

 prove that the limit of u v , where u and v each tend to 0, 

 is always 1 , provided the limit of u -7- -r- v -7- is finite. 



Find the limits of (sin0) tan20 , (tan <9) cos20 , when 6 = fa 

 and of (1 - cos0p 20 when = 0. 



