Let be an open interval of . A real number is called an limit of as x tends to if the following hold: given any real number , there exists some such that
.
Such a , whenever it exists, is unique (see excercise 3 ) and is denoted by .
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Let us look at some examples.
4.1 .7
Examples :
(i) Let if and . Then, . Indeed, .
We find an upper bound for when x is close to 2 , say , that is . Then, .
Thus, given any , we may take and then, .
be an open interval of 
. A real number 
is called an 
limit of 
as 
if the following hold: given any real number 
, there exists some 
such that 
.
, whenever it exists, is unique (see excercise 3 ) and is denoted by 
.
if 
and 
. Then, 
. Indeed, 
.
when 
, that is 
. Then, 
. 
, we may take 
and then, 
. 
