ELISHA MITCHELL SCIENTIFIC SOCIETY. 1 3 



sin X sin x sin x 



. • . lim. = lim. = lim. = i, 



c X tan X 



c c ^ X 



lim. — = lim. ^ lim. ^ i ; 



X tan X tan x 



and it is the same for the reciprocals of the above ratios. 



o 

 Any one of the above ratios approaches the form - indefi- 



o 

 nitely, bnt can never attain it, as the functions cease to 

 exist when x = o and the ratio ceases to exist; but the con- 

 stant value which the ratio approaches indefinitely but 

 never attains {i. e.^ the limit) is at once found to be unity. 

 A function of x is some expression that contains x and 

 is designated by some letter as /J F, ... , with x in paren- 

 theses following. Thus /"(x), F(.r), . . . are read little f 

 function of x^ large F function of .f, etc. If in any func- 

 tion, f(x) of X, the variable x is changed throughout to 

 [x + h) so that the same operations are indicated for (x -f h) 

 as in the original function were indicated for .r, the result 

 is written y(.r -f h). 

 Thus if, 



f{x) = X- cos I — I + log X, 





f(x + Ji) = {x + h) ' cos I ll- log {x 4- //). 



The increase in x (^= //), is called the increment of x and 

 is generally written in the calculus ax, so that h = ^x. 

 The symbol a (delta) indicates a difference, ax signifying 

 the difference between two states of x and the symbol ax is 

 regarded as an indivisible one and not composed of two 

 factors A and x that can ever be dissociated. Similarlv 

 for AjK, A^, etc., when the letters j, 5-, etc., occur in any 

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