When an object is between a converging lens and the focal point of the lens, the three rays appear to diverge. Why is the lens still considered to...

Hello!

A converging lens is called converging because any beam of light becomes less divergent (or more convergent) after going through that lens then it was before.

When an object is beyond the focal plane of a lens, a diverging beam of its rays becomes convergent. When an object is at the focal plane, divergent beam becomes parallel, also less divergent. And in the situation you mentioned a beam remains divergent but less divergent than the initial beam.

The possibility of a lens to compress a beam of light is called optical power. And yes, this power is finite for any given lens.

As an analogy, when some light goes through a sheet of paper, it (light) dims but remains visible. And we may say that a sheet weakens any beam, although complete darkness isn't achieved.