CHAPTER IV. 
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE. 
43. Functions of a positive integral variable. In 
Chapter II. we discussed the notion of a function of a real 
variable x, and illustrated the discussion by a large number of 
examples of such functions. And the reader will remember that 
there was one important particular with regard to which the 
functions which we took as illustrations differed very widely. 
Some were defined for all values of x, some for rational values 
only, some for integral values only, and so on. 
Consider, for example, the following functions; (i) y=x, (ii) y = fix, 
(hi) ?/ = the denominator of x, (iv) y = the square root of the product of the 
numerator and the denominator of x, (v) y = the largest prime factor of x, 
(vi) y — the product of fix and the largest prime factor of x, (vii) y = the 
xih. prime number, (viii) y = the height measured in inches of convict x 
in Dartmoor prison. 
Then the aggregates of values of x for which these functions are defined 
or, as we may say, the fields of definition of the functions, consist of (i) all 
values of x, (ii) all 'positive values of x, (iii) all rational values of x, (iv) all 
positive rational values of x, (v) all integral values of x, (vi), (vii) all positive 
integral values of x, (viii) a certain number of positive integral values of x, 
viz., 1, 2, ..., A, where N is the total number of convicts at Dartmoor at the 
present moment of time*. 
Now let us consider a function, such as (vii) above, which is 
defined for all positive integral values of x and no others. This 
function may be regarded from two slightly different points of 
* In the last case N depends on the time, and convict x, where x has a definite 
value, is a different individual at different moments of time. Thus if we take 
different moments of time into consideration we have a simple example of a 
function y = F(x, t) of two variables, defined for a certain range of values of t, viz. 
from the time of the establishment of Dartmoor prison to the time of its abandon 
ment, and for a certain number of positive integral values of x, this number 
varying with t.