Actually, Amex has the option to set a spending limit for AU's with email alerts if your AU reaches that limit, so you can either keep it or raise it... not exactly what you mentioned, but close.
Quote from Amex help pages:
Spending limits may be set, removed, and modified at any time. You may also freeze an account to suspend spending on it altogether, and you may unfreeze it at any time. The spending limit refreshes each billing cycle
You can assign a spending limit for each of your Additional Cards, giving you more control of your account while giving your Additional Cardmembers the flexibility to enjoy virtually all of the same benefits of your Card
One, you can receive an alert when Additional Cardmember reaches a certain level of spending. You can choose to be notified by text message or e-mail. At that time, you may immediately modify the spending limit