Displays check engine and failure to pick after 3000rpm

You could check the manual book. Check the distances and difference. My Peugeot 408 is at normal level.

limited info -generic answer --sensor failure --improper rebuild --something disconnected -- in limp mode --

You should search BMW forums for your particular year and model to get the best advice. However, it's possible you've got a bad crankshaft or camshaft sensor. At lower RPM, the computer may have a workaround to keep the engine running, but it quits working at higher revs. Borrow a code reader from an auto parts store (free) and follow the instructions to see if a problem has been identified. Buying the part online and installing yourself is usually WAY cheaper than having a shop do it. But check the location of the sensor first, to see if it's a DIY project.

Fuel pump .check fuel pressure

Possibly the new sensor is not the correct one for your vehicle. What is your vehicle, year make, model, engine???

Hi Steve, I suggest having a diagnostic scan carried out on your vehicle. The problem may be related to a defective operation of the Variable Valve Timing or a defective crank or cam pick up sensor. The scan should pin point the problem area without guess work. Regards John

Have the fault codes read. Replace faulty items. Use only genuine items as replacement as non-genuine are not always compatible with the OEM fitted CPU units I suspect an accelerator position sensor or a MAF sensor.

i had the same problem with my pevgeot. its the rev counter getting stuck on and/or sticking xx

Testing TP sensor is easy to test too, just in case it has a dead spot in it.

I don't wish to worry you but I think the turbo kicks in at around 2500rpm and it sounds like it isn't doing so, hence the management light coming on. It could be that failing. Be careful here, apparently the turbo uses the engine oil to maintain its operating temperature and if the turbo seals fail in anyway it eats the engine oil, with the resulting total failure of the power unit. So if you do nothing else, keep an eye on the engine oil level. (You may also notice oil smoke out the exhaust as well).
Unfortunately it looks like a trip to the dealer for them to check what the engine management is saying.