16 VARIABLES AND FUNCTIONS. [INT. II. 



If a function does not approach any limiting value for a 

 certain value of the variable, it must be otherwise denned for such 

 a point ; e.g. we may assign to the function defined at all points 



except x = by the analytical expression sin - any arbitrary value 



x 



for the point # = 0. The function will then be completely de- 

 fined. 



A quantity that approaches the limit zero is called an in- 

 finitesimal. 



If y is a function of x defined in an interval a to b, where b is 

 as large as we please, a number possessing the property that, when 

 M is a given number as large as we please, 



|/(*)-4|<e, x>M, 



for all values of x greater than M, is said to be the limit of y as x 

 increases indefinitely or, briefly, as x approaches infinity. This is 

 denoted as follows : 



= A ; 



e.g. \ime x =l. 



tC=oo 



If in the above definition, we change M to a negative number 

 whose absolute value is as great as we please, and consider all 

 values of x less than M, we say that A is the limit as x approaches 

 minus infinity. 



If in the neighbourhood of a point # = a, when M is any number 

 as great as we please, we can find a corresponding number 8 such 

 that for all values of h, whose absolute value is less than 8, 



\f(a + h)\>M, \h\<S, 



then y is said to become infinite for x = a. If, as above, we change 

 the definition so that y is less than any negative number, y is said 

 to become negatively infinite, or 



lim f(x) = oo . 



The function 



y = sin 

 y x x 



fails to approach any limit, finite or infinite, in the neighbourhood 

 of the point x = 0, by reason of its continued oscillation between 



