Simply add a counter in the while loop:

#!usr/bin/perl
use strict;
open FILE, "<$ARGV[0]"; # $ARGV[0] is the file name  
                      # to pass in in the command line.
my $i=1;
while (<FILE>) {
  if ($i >10) { last;}
  print $_ ;
  $i++;
}
close FILE;
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