Breaking point for SQL server database based on connections from app to SQL server
You can use a load testing tool which supports the protocol which your application servers use for the database communication (OLE DB, ODBC, JDBC, etc.) in order to mimic the increased amount of application servers talking to the database instance, this way you will be able to collect metrics and correlate increasing load with the increasing query execution times, eventual deadlocks and so on. 

Just make sure to configure the load testing tool to use the same connection pattern: connection pool settings, transaction isolation level, etc. so your load test would accurately represent the network footprint of the real application. 

Check out The Real Secret to Building a Database Test Plan With JMeter article for more information on the concept of databases load testing using a performance testing tool, in the above case Apache JMeter