Why rpm goes up and down when gets to last gear..
My car has 208000 miles.
My car has an automatic transmission.
Automatic transmissions can be tricky. When diagnosing them, always start with the simple solutions first. Low transmission fluid can lead to slow and or no shifting between gears.
Check your owners manual for information on checking your vehicles transmission fluid. If you are low, refill and keep an eye on the level. You may have a leak in the system somewhere.
Another component to check without getting incredibly complex is the throttle cable. If the throttle is not responding correctly at certain points, the computer will struggle to demand the correct functions from the transmission.
Doing a scan for diagnostic trouble codes may help guide you to what the issue is. Transmission shift solenoids are another part that may cause these types of symptoms, but come with other symptoms you would have noticed as well.
If you need help with this, I would suggest having a certified technician, like one from YourMechanic, diagnose and service your vehicle at your home or office for an accurate repair.