How can i tell if the fuel injector is dead?

plug in a noid light, see if its getting a signal. If so, listen for the injector opening and closing with a stethoscope, if it isn't, replace it. This could also be an ignition problem, or low compression on that cylinder.

make sure the fuel injector has power when the car is running, and check the other wire for ground ( a flashing ground when running, as this is controlled by the computer) If no ground , then problem may be in wire , or the computer driver for that injector. Listen for any drop in rpm when unplugging the suspect injector, it will make the car run noticeably rougher if it is ok, if no drop in rpm, that is the cylinder to look at. Make sure there is a good spark to that cylinder, and no vacuum leaks around that cylinder area.