If you get 0/0 from your substitution, you most likely have not understood what you are doing. If substitution is to have any value, you must FIRST demonstrate continuity. You must not, therefore, wonder if your result from substitution might imply a discontinuity. Your thinking is backwards.
Please go back and study the definition of a limit.
After that, look up and try to comprehend "Indeterminate Forms".
If you choose to substitute wily-nilly, your existing, finite values are very encouraging, but they are not conclusive.