(ii) Let , the greatest integer function. Clearly, for and for
Thus, if we take a sequence , then clearly, and , as . On other hand, if we take sequence , then again , but , as . Thus, we cannot predict a single value for at .
Click here to see an interactive visualization: Applet 2.2
This motivates the following definition.
4.1 .2
Definition :
Let be an open interval of . We say that has limit at if there is a real number with the property that , for every sequence with .
Such is unique (see exercise 3), whenever it exists and is denoted by .
In view of the algebra of limits for sequences (see section 3.2), we have the following theorems.
, the greatest integer function. Clearly, 
for 
and 
for 
, then clearly, 
and 
, as 
. On other hand, if we take sequence 
, then again 
, as 
. Thus, we cannot predict a single value for 
at 
.
be an open interval of 
. We say that 
if there is a real number 
with the property that 
, for every sequence 
with 
. 
. 
