The first signs of battery damage are going to be that the device does runs out of battery power with 2 hours on a full charge, or the device is intermittently powering off.
Other physical signs of a swelling battery will begin to show also. If you can squeeze your fingernail anywhere between the seam where the white and black plastic meets, you are likely experiencing a swelling battery. In this picture, I have already removed the back, which you should not do if you have a swelling battery, but it is the best picture to show you the slight gap between white and black plastic that formed as a result of the swollen battery.
Case with a swollen battery:
Normal Case
