MEANING OF THE WORD INFINITE. o 



obtained in Art. 7, namely that the limit of - when x 



L -f- x 



increases indefinitely is unity, would be sometimes expressed 

 thus, "when x is infinite - - equals unity." Here both 



parts of the sentence are abbreviations: "when x is infinite" 

 can only be considered as meaning " when x is increased 



SO 00 



without limit," and "- equals unity" means strictly "- 



can be made to differ from unity by as small a quantity 

 as we please." 



CO 



11. In the example 11 = let us now ascribe to x 



J x 4- I 



g 



negative values. Put z for x ; thus y = . Now sup- 



Z I 



pose z to change gradually from to 1 ; the numerator of y 

 is positive and continually increasing, while the denominator 

 is negative and numerically continually diminishing. The 

 value of y then is negative and numerically continually in- 

 creases, and by taking z sufficiently near to unity we may 

 make y as great as lue please ; that is, as z approaches unity 

 y has no finite limit. For the sake of shortness, this is some- 

 times expressed thus, " y is infinite, when z = 1 ;" but it must 

 not be forgotten that this last phrase is an abbreviation, and 

 must be considered to mean : " by taking z sufficiently near 

 to unity y can be made to exceed any assigned magnitude, 

 however great." We shall not proceed further with the ex- 

 ample ; the reader will see that when z is greater than unity 

 y is positive, that y continually diminishes as z increases, and 

 approaches the limit unity when z increases indefinitely. 



12. The student has already seen an example of the same 

 kind as that brought forward in the last Article, for he has 

 probably been accustomed to say, " the tangent of an angle 

 of 90 is infinity." On reflexion he will see that the only 

 way in which a meaning can be given to this statement is 

 to consider it an abbreviation of the following : " as we 

 increase an angle gradually up to 90, the tangent of the 

 angle increases, and by taking the angle near enough to 90 

 we may make the tangent as great as we please." We can 



