A price of tartar i'm guessing broke off my front lower teeth while eating and now the area is exposed and pink like a pocket kinda between the gum and the tooth without pain it's just sharp on my tongue can tartar break off or is it my actual tooth?

Hard deposits can break off. If you have that much calculus then you are way overdue for a professional cleaning by a dentist. Left untreated, the bacteria residing in those deposits cause gum recession, bone loss, and eventual loss of teeth. Make an appointment with your dentist.

Probably tarter. As you have seen, these hard bacteria collecting deposits cause gum irritation, infection, recession, bone loss, eventual tooth loss. They accumulate when teeth improperly brushed and especially not flossed. You can't remove them yourself. Please, see a Dentist. Your gum disease will not improve without professional care.