Rob, I have a gremlin that I really don't want to chase like I used to. My passenger-side lamp is out and it is not the bulb or the number 2 or number 11 fuse. Is there a relay somewhere that, after 25 years, has finally given up the silicone ghost? This old lady has 223K on the clock and is still pretty much OK. The AM radio still sucks w/o a real antenna. Just never felt like taking the drill to the deck lid (old tii black pull up job might look the part). #4480 in Connecticut
Have you checked power at the connector for the bulb that is out to see if this is a power or ground issue?
I only had one E34, it was many years ago, and I didn't really tear into the electrical issues. Still, this site reports that it does have relays for both low and high beams: http://www.bmwe34.net/E34main/Maintenance/Electrical/LKM.htm
Another common issue on E34's is the LKM - rather than tackle working on mine, I lucked into somebody selling a Programa-rebuilt one on ebay; You can try some of the repair procedures if you narrow it down to the LKM, or source a replacement from Programa See Rob's link, above, also: http://www.bmw4life.com/lkm.htm Start at the beginning and input the correct model, year, etc. for your car here: http://www.programainc.com/categories.aspx?idcategory=3041&cl=1