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1. First of all, get the battery charged up to 100%. Every BMW owner should own a good smart battery charger (or battery maintainer). These usually put out between 0.75 amps to 1.5 amps and you'll need leave it on at least 8 hours. I suspect you will need a new battery but at least getting your present one up to 100% should provide some useful information (ie: is the car running better now?). Yes, a bad battery can cause a multitude of funny issues.
2. Fan running fast & limp mode usually indicate a failed water pump.
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