Today we migrated to AzureSQL V12. Tonight my site is offline, because of a persistent issue with the following message:
Resource ID : 3. The LoginLimit l
This issue was caused by Azure automated backups failing to end sessions correctly. None of the other answers worked as they all require you to be able to connect to the database, and I was unable to do this. I ended up speaking on the phone to Microsoft Support for some hours, during which they were also unable to connect to the database for the same reason.
The resolution, from their end, was to migrate the database to a new node, which is not something that Azure users can do, so if you encounter this level of issue, really the best (and only) thing to do is to contact Microsoft support ASAP.
I do recommend trying out some of the other suggestions here first, but without the ability to make any connections you will be stuck.
Because this occurred during the process of upgrading a database on Azure, we implemented a procedure of disabling automated backups before upgrading databases as a precaution, and the problem has not reoccurred for us.