The antenna amplifier / receiver in the tailgate is a known fault.

Multiple buttons involved in pairing so any one malfunctions and you're sol.

Some hardware stores have a remote tester that will blink when it senses a
Signal so that's a way to test if the buttons are doing anything.

If you poke a pair of needles into the key in exactly the right place you can measure the voltage of battery. It will drop a little when you press a button if it drops a lot battery is shot.

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2011 E70 • N55 (me)
2012 E70 • N63 (wife)