A/C won't work off of the batteries and inverter at all or for very long depending on how powerful of inverter you have. Generator for the A/C.
We have two 12v deep cycle batteries and will be able to run a couple of 12v lights, a sat receiver (or DVD player) and TV off of the inverter for about 6 hours. This includes all of the usual 12v draw items like refrigerator control unit, carbon monoxide detector, and water pump.
If we turn on the heater then the time drops to about 4 hours. Depending on how cold it is and how long the fan runs to maitain a comfortable temp inside.