Only way to determine if .the fan is the problem is to get it scan. A bad fan will give a DTC code, but will not set off the SES or CEL since it is not related to the engine. I use BMW Scanner 1.4 and when my fan went bad I got a DTC code for the fan. Try having the software turn on the fan but no go. Changed the fan and every thing was cleared and the AC was blowing ice cold again.

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2006 Infiniti G35
2001 BMW 3.0I E53 X5 Build date 08/2000 SOLD
Lotus Europa 1970 Destroyed by fire
Lotus Europa 1970 S2 Renault Powered
Lotus Type 52 1970 Twincam Webers Powered
PORSCHE 911 Targa 1982 The Garage Queen
Audi Avant donated to Kars for Kids
BMW 525IT Sold
Audi 4000CS Quattro Sold
Jensen Healey Lotus Powered Sold
Opel 1900 Sold
Triumph Spitfire 1971 Sold
Triumph Spitfire 1968 Sold
Plymouth "Cuda" 340 Six pack SOLD