The easiest query for you to run for checking a SQL Server for a particular database is to query the
MasterDB for a count of databases matching that name
SELECT Count(*) FROM [Master].Sys.Databases WHERE [Name] = 'Person'
If this returns a value of 1, you can then query that
PersonDB for tables matching the one that you want
SELECT Count(*) FROM Person.Information_Schema.Tables WHERE [Table_Name] = 'PersonInformation'