I have fault codes 5eb3 and 5f3a on a e53 x5 3.0d can you help

If you take your registration ducument to a BMW dealership they will tell you the code for free, based on the VIN number.

Or look in the Glovebox for a small white sticker, or in the Owner's manual.

But, I found this

BMW hasn't used a "radio code" since the beginning of the e38/e39/e53/e46 generation of BMWs. That technology ended with the anti-theft radios used in BMWs prior to the aforementioned e series.

The code you see appearing on your nav screen or MID (you didn't indicate which radio setup you have)...is an anomaly of the OBC's immobilizing code probably brought on by the low voltage issues you were having. There have been many threads on this topic across multiple BMW forums in that usually a simple use of "0000" or "1234" works to get past the lockout once the battery issue or low voltage issue is fixed...or initiating the emergency deactivation of the CODE found in the owner's manual.

and this confirms it is the immobilizer code

https://www.bimmerforums.com/forum/showthread.php?2270312-radio-code

..

why not u try a different battery?

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You can't reset a fault, until the fault is repaired that is causing the fail safe mode the problem will persist. You need the engine and or the Transmission control module scanned for fault codes.

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

Diffen Gunari

A scanner can reset any light on the dash, if the problem has been fixed. A code reader will reset the engine light, again if the problem is fixed.

failed electronic stability control module.sits behind your gear stick.take it to a dealer to check the codes first.as they can check that part first. as two other things can go wrong in your gear box .but the codes will check that.

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

5eb3 is a DSC-sensor power supply and the 5f3a is a transfer case malfunction. hope this helps