MISCELLANEOUS EXAMPLES. 



223 



MISCELLANEOUS EXAMPLES. 



1. Shew that if x and c are positive 



_, x c c 



2 log -- h - H 







X C + X 



decreases as x increases. 

 2. Shew that if x and c are positive 



c-+xj 

 increases as x increases. 



3. If u=(x 3}e ix +4:xe x +x+3 shew that -j 2 , -7- , and u 



d/x dx 



are positive for all positive values of x. See Ex. 10, p. 86. 



4. Shew that for positive values of x the expression 



e 2 * (x - 2) + e* (x + 2) 



(**-l) 3 

 diminishes as x increases, and that its greatest value 



is I- 



5. Demonstrate the following approximate expression when 



x is small, 



x 



6. Evaluate 



(1 + X ^ ~ 



x 



when x = 0. 







Result. - . 



z 



7. Shew that when x is infinite 



8. Find the value when,# is infinite of 



8ic 2 ( 1 + -Y - 8ex 3 log f 1- + i) . 

 \ */ & \ a?/ 



Result, e. 



