G DEFINITION OF A LIMIT. 



form no distinct conception of an infinite magnitude, and the 

 word can only be used in Mathematics as an abbreviation 

 in the manner of the examples here given. 



If to x the independent variable be ascribed values begin- 

 ning with zero and increasing without limit, this is sometimes 

 expressed for abbreviation by saying that x increases from 

 zero to infinity. 



13. The meaning of the word " limit," or its equivalent 

 "limiting value," will be understood from its use in the 

 preceding Articles. The following may be given as a defini- 

 tion : " The limit of a function for an assigned value of 

 the independent variable, is that value from which the 

 function can be made to differ as little as we please, by 

 making the independent variable approach its assigned 

 value." 



14. In the example " the limit of - = 1 when = 0," it 

 is obvious that ^- is never equal to 1 so long as 6 has 



V 



any value different from zero, and if we actually make 



sin 6 

 = 0, we render the expression - unmeaning. In other 



words, although ^ approaches as nearly as we please to 



the limit unity it never actually attains that limit. Some- 

 times in the definition of a limit the words " that value 

 which the function never actually attains" have been in- 

 troduced. But it is more convenient to omit them ; for if 



/* 



we take any function of x, say , and ascribe to x any 



X *T" i 



value, say 1, we can determine the actual value of the 

 function, which in this case would be |. According to the 

 definition we have given in the preceding Article we may 



/v* 



if we please call \ the limit of when x approaches unity. 



SC "^ L 



The same holds for any finite value of any function, and 

 generally according to the definition of a limit laid down 

 in Art. 13, any actual value of a function may be considered 

 as a limiting value. 



