I've been racking my brain on this problem and I've got the possible solutions down to 2 but I can't seem to eliminate either one definitively. Here is the problem
Using the "limiting case" recommendation from the question, I figure that we can eliminate (D) and (E) because for either of these equations to be true, it must be the case that wavelength is not dependent on the mass of the string or the mass of the ring, which couldn't be true.
For (A), (B) and (C), when , and the sine component to (C) will just keep it oscillating between its amplitude, so we can eliminate that.
However, I can't for the life of me eliminate (A) or (B). I considered the possibility that only certain wavelengths would be allowed when you have a fixed endpoint, which is technically true, and so I argued that with the fundamental (first harmonic) wave with just two nodes, you would have the , at which point the Cot solution would go to infinity and Tan would go to 0. In this case the Tan solution would be correct. However, if the wavelength = 1/2 L, then the opposite is true. It seems to me the only way that ONLY (B) can be the solution would be if letting the mass of the ring go to infinity would force . This doesn't make sense to me because I know you can create multiple modes for a standing wave even with 1 or both of the endpoints fixed. However, I'm wondering if maybe the problem is trying to suggest, by telling us that the string has a tension T, that the cord is pulled tight. Under this condition I might be willing to accept that the wavelength is restricted to only a wave with 2 nodes.