I had a dentist inject my mouth back on 12/11/16. My right side of my tongue and bottom under my tongue is still partially numb. Could I have a case?

There is a definite possibility that you have a dental malpractice case. It depends on whether another dentist will testify that your dentist deviated from the accepted standard of dental care by doing or failing to do something, and that but for that error, the dentist is reasonably certain you would not have sustained this numbness.