Final answer:
The hippocampus within the limbic system is most closely associated with memory formation and impairment to it would likely result in memory problems.
Step-by-step explanation:
Damage to the hippocampus would most likely impair memory within the limbic system. The limbic system is integral to emotional response, behavior regulation, and especially memory formation. The limbic system includes the hippocampus, amygdala, parts of the thalamus and hypothalamus, and the cingulate gyrus. The hippocampus, located in the medial temporal lobe, is particularly important for the processing of explicit memories. Impairments to this area can lead to difficulty in forming new memories, as evidenced by the famous case of patient H.M., who lost the ability to form new semantic knowledge after his hippocampi were removed. In contrast, the cerebellum is responsible for motor learning and the conditioning of reflexes, not explicit memory formation.