As runrig said, syntaxes can vary... so here are some common ones:

PostgreSQL, MySQL, and some other have variants of the LIMIT x OFFSET y clause. This always (to my knowledge) appears as the last clause in the query.

...
ORDER BY field1, field2
LIMIT 10

-- with OFFSET

LIMIT 10 OFFSET 1

-- implicit OFFSET (MySQL)

LIMIT 10,1
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In Oracle you can use the "rownum" indexing of the recordset to return a specific number of records.

...
FROM table1 a, table2 b
WHERE rownum <= 10
...
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If you're working with an M$ DBMS like SQL Server (I refuse to allow Access to be called a DBMS) or one Informix's you can use the TOP syntax. TOP is part of the SELECT clause as seen below.

SELECT TOP 10 field1, field2, ... ,fieldn
FROM ...
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I know nothing of DBI so the next statement is a generality. The benefits to using the limiting at the DBMS side are smaller recordsets transfered back and often quicker queries (DBMS dependant), you do loose a certain amount of portability however. Staying with straight ANSI SQL and datatypes is always a problem though.

Just have your fetch loop exit after 11 rows are fetched. That will give you the 11 oldest rows. It shouldn't matter if there are fewer than 11 rows, then the loop will end without an explicit exit. Queries can sometimes be limited to a certain number of rows, but it is database dependent and the syntax varies.

You want to select the oldest 11 rows (after a certain date), without ordering by date and time... But if you don't order by date and time, how do you know which are the oldest??

To fill out your code a little more:

#!perl -w

use strict;

use DBI;  # note: must be uppercase

my $dbh = DBI->connect("dbi:ODBC:burlee", "username", "password")
  or die "Can't connect to database: $DBI::errstr\n";

{

# $sth is a static variable for get_sport()
# it's private to get_sport(), and keeps its value between calls
my $sth;

# get_sport($sport, $import)
# returns a list of array references
sub get_sport {

  my ($sport, $import) = @_;

  my $path = "d:/text/time/$sport.txt";

  open(TEXT, $path) or die "Can't open $path: $!\n";

  my $date = <TEXT>;
  chomp $date;

  close(TEXT);

  # prepare, if it wasn't prepared before
  $sth ||= $dbh->prepare(<<"  EndOfSQL");
    select *
    from DATA_DB_ENTRY__ASTROS_STAGING
    where TS_DATE >= ?
    and IMPORTACE = ?
    order by TS_DATE asc, TS_TIME asc
  EndOfSQL

  $sth->execute($date, $import);

  my @rows;

  while (my(@row) = $sth->fetchrow_array()) {

    push @rows, [@row];

    if (@rows == 11) {
      $sth->finish();
      last;
    }

  }

  return @rows;

}

}
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set rowcount 11

select
  first_name
from
  database
where
  start_date <= '2001/04/14'
  and stop_date > '2999/12/31'
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