I may not be understanding but it sounds like, if any person has a date of 1/1/1900 you don't want them, which should eliminate Jan Klass correct?
For the sake of this example, I'm ignoring the ID column since it appears to not matter in the example you've put forth.
DECLARE @ExitTable TABLE (
FirstName varchar(200) NULL,
LastName VARCHAR(25) NULL,
Exit_Date DATETIME NULL
);
INSERT INTO @ExitTable( FirstName, LastName, Exit_Date )
VALUES ( 'Jan',
'Klass',
'1-1-1900'
)
INSERT INTO @ExitTable( FirstName, LastName, Exit_Date )
VALUES ( 'Jan',
'Klass',
'1-1-1900'
)
INSERT INTO @ExitTable( FirstName, LastName, Exit_Date )
VALUES ( 'Jan',
'Klass',
'12-25-2017'
)
INSERT INTO @ExitTable( FirstName, LastName, Exit_Date )
VALUES ( 'Peter',
'Jobs',
'1-1-1900'
)
INSERT INTO @ExitTable( FirstName, LastName, Exit_Date )
VALUES ( 'Mark',
'Mulder',
'6/24/2017'
)
INSERT INTO @ExitTable( FirstName, LastName, Exit_Date )
VALUES ( 'Taylor',
'Mulder',
'11/30/2017'
)
INSERT INTO @ExitTable( FirstName, LastName, Exit_Date )
VALUES ( 'Mike',
'Mulder',
'8/30/2017'
)
INSERT INTO @ExitTable( FirstName, LastName, Exit_Date )
VALUES ( 'Big',
'AL',
'1-1-2017'
)
INSERT INTO @ExitTable( FirstName, LastName, Exit_Date )
VALUES ( 'Big',
'AL',
'10/30/2017'
)
INSERT INTO @ExitTable( FirstName, LastName, Exit_Date )
VALUES ( 'Big',
'AL',
'1/5/2017'
)
SELECT * FROM @ExitTable AS A WHERE
CAST(A.Exit_Date AS DATE) > CAST('7/29/2017' AS DATE)
AND NOT EXISTS (SELECT * FROM @ExitTable AS B WHERE CAST(B.Exit_Date AS DATE) = CAST('1/1/1900' AS DATE) AND B.FirstName = A.FirstName AND B.LastName = A.LastName)
The NOT EXISTS eliminates any record where they may have an entry of 1/1/1900
Expected output should be
Taylor Mulder 2017-11-30 00:00:00.000
Mike Mulder 2017-08-30 00:00:00.000
Big AL 2017-10-30 00:00:00.000