First, we note that the function is defined at with , so (i) and (iii) are not correct.
Now let’s see whether the limit of the function at exists. For this purpose, we consider the left limit and the right limit. When approaches to from the right, we consider points near which is greater than . For such points, the corresponding piece of the function is , so that
Here we use the given information
.
On the other hand, to get the left limit, we consider points close to . Since the corresponding piece of the function in such region is , we get
Since the left limit is not equal to right limit at ,
the limit
does not exist. In all, we choose
.