16 DIFFERENTIAL AND INTEGRAL CALCULUS. 



( 1\* 



3. Prove that the limit of 1 1 4 1 when x is increased 



V xj 



indefinitely is the limit of the sum of the series 



and that the corresponding limit of ( 1 4 -] is that of the 



\ xj 



sum of the series 1 4 a4p- ++] K.. to oc, and is 

 also s a . 



4. Deduce the limits of -log a (l+2) and of , 



when z is indefinitely diminished. 

 Prove that the limit of the product 



2 2 2 2 



when n is indefinitely increased, is . 



a I 



5. Define the differential coefficient of a function of any 

 independent variable. Give a geometrical illustration of 

 the meaning of a differential coefficient, and prove that if 

 $ (x) and cj)' (x) be finite and continuous, 



where 6 is some proper fraction. Obtain the differential 

 coefficients of \f(x)^ (a + x), and \/(a 2 x 2 ). 



6. Obtain the differential coefficients of 



cc n , a*, e*, a+0a; , log a a;, and log {x 4 V(l 4 x 2 ). 

 1. Differentiate 



since, cosa?, tan#, sin 3 a;, log tan (ITT + Jo?), log tan ^a;, 

 sin" 1 (sin a? cos a-), and log{sina; + coscc + V(siu2ic)}. 



8. Prove the rule for differentiating any product, or 

 quotient, of different functions of x. Differentiate the pro- 

 duct (1 + x + x 2 ) (1 x + x 2 ) (1 # 2 + # 4 ), simplifying the 

 result. 



