It depends on the details of your treatment of limits.
There is no emperor and no controlling council for mathematical terminology and notation.
In some treatments of limits,
In this way of defining limit, you function does not have a limit at
In other treatments , limits at endpoints of closed interval domains are defined to be the one-sided limit (from the appropriate side.)
In this treatment, is is possible that the function has a limit.
In the first treatment, we have
So the second treatment allows