Hopefully the bills are all in the tenant's name.
If not you need to get all the meters read, i.e. gas, electric, etc., not in the tenants name, and have them transferred to yours as of the date they vactate.
That way you can determine where they stopped and you started.
Be sure and get a forwarding address for them and just send their bills along after they vacate.
You can also ask them to fill out a change of address form at the post office.