8, 9] VARIABLES AND FUNCTIONS. 15 



positive number e, however small, there may be found a corre- 

 sponding number 8 such that for all values of h of absolute value 

 less than B, 



+ h)-A\< , \h\<S, 



then the function is said to approach or converge to the limit A 

 in the neighbourhood of the point x = a. This will be denoted by 

 the equation 



=A. 



The necessary and sufficient condition for the existence of a limit 

 at a is that 



\f(a + h)-f(a + h') \<e, 



\h\<S, \h'\<8, 



where e, 8 have the same significations as before, and h and h' are 

 any values whose absolute values are less than S. If the above 

 condition is satisfied only when h and h' are positive, the function 

 is said to approach the limit on the right of a, if when h and h' are 

 both negative, on the left. A function may approach different 

 limits on the two sides of a point. It is not necessary that a function 

 should be varying always in the same sense in order to approach a 

 limit, e.g. the function 



y = x sin x, 



which alternately increases and decreases, approaches the value 

 zero as a limit in the neighbourhood of x 0. The function 



approaches the limit zero on the right of x = 0, but not on 

 the left. 



The function 



. 1 



y sin - 

 x 



does not approach any limit whatever in the neighbourhood of 

 x = 0, for in any interval, however small, from 



2 2 



X = ; - to X = -TI ?rr > 



where n is any integer, however great, the function takes all 

 values from 1 to 1. 



