Final answer:
The best prognosis for a tooth with a broken instrument close to the apex is when the tooth has a vital pulp without a periapical lesion, as the healthy pulp can respond to infection or injury.
Step-by-step explanation:
The prognosis of a tooth with a broken instrument located 3 mm from the apex depends significantly on the vitality and health of the pulp and surrounding periapical tissue. Among the options provided, the best prognosis is likely with B. vital pulp without a periapical lesion.
A vital pulp indicates that the pulp tissue inside the tooth is alive and healthy and can mount an immune response to the foreign body, which can aid in recovery or containment of any infection. Conversely, a tooth with a necrotic pulp lacks this capability as the pulp is dead, and the presence of a periapical lesion indicates existing inflammation or infection, which could complicate or hamper healing and may necessitate further endodontic treatment or surgical intervention.