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If there is no written lease that is in force for a fixed term, then the lease defaults to a month to month tenancy by law. Either party can terminate the tenancy with proper notice as required by state law. In IL, either party can terminate with a written 30 day notice delivered to the other party specifying the date on which the tenancy will end.
After giving notice, the landlord would then only be able to demand rent for the next 30 days and the tenancy would then terminate.
But if the tenant doesn't give a 30 day notice, the landlord could sue them and hold them liable for up to one more month's rent under a breach of contract claim.
Correct. Absent any specific requirement in the lease that notice has be given prior to the first of the month to be effective on the 1st, the 30 day starts running the day after it is delivered to the landlord or tenant with the tenant being liable for rent for the entire notice period. If it included some time that had already been paid for, then any remaining rent for the next month should be prorated.