The battery may not be completely dead; it is possible for it to have a problem that lets it put out normal voltage for a light load, but not be able to crank the engine. Pray that that's so, because that's about as inexpensive a problem as you can have.
How old is your home A/C? They're usually good for 10-15 years...
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How old is your home A/C? They're usually good for 10-15 years...