BMW Aux Fan Resistor ↗
Diagnosing a dead two-speed electric auxiliary fan. It lists the usual culprits in order (the radiator's double temperature switch, the cigar-shaped fan resistor, the low and high-speed relays, the motor, the fuse) and explains how the resistor gives low speed. Tests include measuring the switch across its pins and checking 12 V on the blue and black wires before you fit a new resistor.
Tools mentioned: Multimeter, jumper wire, screwdrivers
Written around a 318ti (M42) but says it applies to all models with an aux fan and resistor, incl. Z3.