This is a question regarding the connection between Fargate and RDS.

Hello.

If the Fargate task disappears when the application process ends, the connection to RDS will disappear.
In other words, I think that a connection to the database is created when the task starts (when the application starts processing).
So, if you are having problems with the number of connections to the database, you can improve it by setup something like RDS Proxy and using a connection pool.
https://docs.aws.amazon.com/AmazonRDS/latest/UserGuide/rds-proxy.html