I guess you are talking about limits like in analysis, but even then I don't think this is really true. I'm not really sure if you intend to distinguish between

I guess you are talking about limits like in analysis, but even then I don't think this is really true. I'm not really sure if you intend to distinguish between limits and colimits; I'll assume not, but I would certainly agree that infinite (inverse) limits are harder to understand with diagrams. On a simple level, limits in category theory don't need to involve infinite things at all, you can take the limit over a small category like (x --> x <-- x). With regard to derived concepts, one can draw pictures of "projective resolutions" or draw level by level pictures of the nerve of the category you want to work with. So I think you can in fact make useful pictures even in this case.