Don't believe the dealer, they want to rip you off.
In 99% of the time, it is the FSR (happened to my E39 1998 528i).
Get a new FSR from dealer ($90) and stick it in there.
Or get a good used OEM FSR on ebay for $20 and stick it in there.
Or junk yard FSR.
If you want to troubleshoot the FSR circuit, I posted info for E39 below:
DIY: Trouble-shooting HVAC Blower Motor and FSR - Bimmerfest - BMW Forums