What is the way for the client to be able to find out which name server has information about the specific domain name that the client is interested in?

The client begins by querying NS1 which is root name server and asking for the A record for say, web.example.com. NS1 does not have any information about web.example.com or even example.com, but it does have a name server record stating that the com zone is stored on NS2. NS1 sends a response to the client containing that name server record. Essentially NS1 says to the client, I don't have an A record for the domain name web.example.com, but NS2 does hold the com zone, so go ask NS2. The client then queries NS2. NS2 looks at its database and sees that the example.com zone is stored on NS3. So it sends the client that name server record that says NS3 holds the example.com zone, so go ask NS3. The client then asks NS3 for the A record for web.example.com. NS3 looks at the database and it sends that record, which contains the resource record data, that is the IP address say, 1. 2. 3. 4. This process here is called recursion. The clients keep following this trail of name servers until it either locates the record it's looking for or it gets a response indicating that the record doesn't exist.