Classifying divergent sequences in a metric space

Question

To my understanding, in $\mathbb{R}$, we have the following ways in which a sequence can diverge:

1. The sequence could diverge off into $\infty$ or $-\infty$ (relevant generalization)

2. Divergence by oscillation: A sequence that neither converges to a finite number nor diverges to either $∞$ or $−∞$ is said to oscillate or diverge by oscillation.

   2.1.An oscillating sequence with finite amplitude is called a finitely oscillating sequence. eg: $\{ (-1)^n \}$

   2.2.An oscillating sequence with infinite amplitude is called an infinitely oscillating sequence. Eg: $\{ n(-1)^n \}$

In contrast, for a general metric space, is there a way to classify the ways of divergence similar to the above? If so, what are some examples of ways which a sequence could diverge in a general metric space which can't happen in $\mathbb{R}$?

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Notes:

On divergence by oscillation Source