EXAMPLES OF A LIMIT. 3 



6. Explicit functions may be divided into algebraical and 

 transcendental. The former are those in which the only 

 operations indicated are addition, subtraction, multiplication, 

 division, and the raising of a quantity to a known power 

 or the extraction of a known root; the latter are those which 

 involve other operations, as exponential functions, logarithmic 

 functions, and trigonometrical functions. We suppose here 

 that the number of the operations indicated is finite ; for as 

 we shall see hereafter a transcendental function may be equi- 

 valent to an infinite series of algebraical functions. 



To the independent variable in an equation we may 

 suppose any value assigned, either positive or negative, as 

 great as we please or as small as we please. If we suppose 

 a series of different values assigned to x, beginning with 

 some negative value numerically very large and gradually 

 increasing algebraically up to some large positive value, 

 the series of values we obtain for ?/ may present to us very 

 different results. For example, if y x 3 , then the values 

 of ;/ will form a series beginning with a negative value 

 numerically large, and increasing algebraically up to a large 

 positive value. If y = x\ the values of y are always positive, 

 and form a series first decreasing and then again increasing. 

 If y = \/(a 2 x'~), then the values of y are unreal for every 

 value of x not contained between a and + a. 



7. We proceed to another example more important for 



cc 

 our purpose. Suppose y = , and consider the series of 



-L ~T~ 3C 



values which y assumes when to x are assigned different 

 positive values. When x = 0, y = 0, and when x has any 

 positive value, y is a positive proper fraction. If we 



put y in the form 1 , we see that as x increases 



so does y, but y being a proper fraction can never be so 

 great as unity. The difference of y from unity is ; 



this fraction diminishes as x increases, and can be made 

 smaller than any assigned fraction, however small, by 

 giving a sufficiently great value to x. Thus if we wish 



y to differ from unity by a quantity less than , 



100,000 



B2 



