Module 2 : Limits and Continuity of  Functions
Lecture 4 : Limit at a point
4.1 .10
Definitions :
 
Let be an open interval, . Let.
(i)
We say has left-hand limit at a point , if there is a real number with the property that for
 
every ,  there is some    such that
                                                         .
We write this as , and call to be the left-hand limit ofat.
(ii)

We say a functionhas right-hand limit at a point x =c if there is a real number with the property that
  for every there is some such that

                                             .
     We write this as , and call to be the right-hand limit ofat.
     The above remarks tell us the following :
4.1 .11 Theorem :
  Let and be such that  is contained in * for some *.Then exists and is equal to if and only if as well as . That is the limit of a function at a point exists and is equal to if and only if both, the left- hand and the right hand limits exist and are equal to .
4.1 .12 Examples :
(i) If , the greatest integer function, then
                                            
Thus, does not exist.
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