Why is my E53 BMW 3.0d cuts off while driving.

On a 2001, and because it is only on one side of the vehicle, and because it is the drivers side only, my guess is that the wire that goes from the driver's door to the door frame is damaged. The best course of action would be to take it to an auto electrician for diagnosis with their pro grade scan tool computer. There are some other theories in the link.

https://www.google.com/search?q=2001+BMW+X5+e53+left+side+front+and+rear+doors+will+not+unlock

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run fault codes and check foe crank/cam shaft position sensor problems
voltage is good ( 14.5-14.8 max)

Fuse box is located behind glove box and check fuses No. 26, 32 and 33

https://www.youtube.com/watch?v=rQn5J47ElwU

Check online BMW Workshop Manuals for X Series E53 X5 3.0d (M57) in the attachment: http://workshop-manuals.com/bmw/x_series_e53/x5_3.0d_m57_offrd/2_repair_instructions/12__engine_electrical_system_(m57)/21__preheating_relay/1_ra__removing_and_installing_replacing_relay_for_glow_plugs_(m57)/

A combination of items, transfer case chain, electronic actuator or VTG control unit, have you scanned it yet

Diffen Gunari

for better result download the pdf file
1999 2006 Bmw X5 E53 Workshop Service Repair Manual

Hi Mary,here's a link to your firing order. Specifications Firing Order AutoZone com

2003 X5 bmw the trans. fail safe program comes on when driving, I was told that the trans needs to clean out and put new sensors. Is that true it gives the code 055

hi Andrew, try to press the pairing button and keeps hold for 3 sec. then switch the ignition in first position while still holding the button for another 3 sec then the the pairing messeage should be sig on. but it the pairing messeage did not show i suspect that your ULF control unit is faulty and needs to be replace.