I am not a FL lawyer, but the general rule is that owner of trees are only responsible for falling limbs (or falling trees) if they have prior notice of a dangerous condition of the tree. I can't resist mentioning that a neighbor's tree once fell on my car in my driveway and flattened it. The tree was dead and was obviously rotten before it fell. My auto insurance paid for the car. His insurance company would not pay a dime and my insurance company did not even bother to try to collect from them. If I had sued him in small claims court I could probably have collected for my deductible, but it wasn't worth spending the time and riling up a new neighbor.
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I agree with Mr. Brophy concerning the general rule of liability for falling trees. However, why speculate? Tender the claim to your homeowner's insurance company. They will either pay the claim or pay to defend you and then, if they lose, pay the claim. The greatest risk you can take is trying to sort through this yourself and stepping in a bucket that your insurance company would deftly evade - if you settle or negotiate your tenant's claim before your insurance company has a chance to investigate and negotiate it themselves, they can resist paying your claim and leave you out in the cold.
Take photos of the tree before too much time passes, as well as photos of limb itself, core of limb would be best. Helps to determine if there was some disease process one should have recognized or whether a healthy limb broke due to storm. Provide this to your homeowners carrier if you have insurance. Before telling tenant to buzz off, ask yourself how good a tenant he or she is and how hard would it be to find a new tenant, vs cost of the damage. If you throw this to carrier and they deny claim, which will likely occur, and tenant is mad at you and moves, you may lose 750 per month for 4 months while looking for a new renter. Thats a 3000 loss. If damage is say 1,000, it might be wise to try and help out a bit with the repair instead of losing the renter. Just weigh all factors, money wise, before deciding how to move forward. But bottom line, from a legal standpoint, you probably don't owe damages....there is sometimes more to consider than just the right legal answer however.