LLC and Assumed Names

Our bank has a new customer that is organized as an LLC. The LLC is conducting business under a separate name and the checks are made payable to this business name. I have read through the Assumed Business Name Act (805 ILCS 405/) and found that LLCs are exempt from this act. However, does this mean they are exempt if they operate under the LLC name, or are they exempt completely? I can't see how they can be completely exempt because how would our bank be able to prove that we can accept checks made payable to a business that is completely different from the name of the LLC? I would think we would need something (i.e. assumed name certificate) showing the LLC operates its business under the separate business name.

Thank you for your help.

An LLC's use of a trade name is governed by 805 ILCS 180/1-10.