The LIMIT 3 should be fine, because if the offset isn't defined, it defaults to 0.
I am wondering why you are escaping the double quotes in the sql though..
Change this:
$sql = "select * from article where permanent_submit = \"yes\" AND hot_topic = \"yes\" ORDER BY article_date DESC LIMIT 3";
to this:
$sql = "select * from article where permanent_submit = 'yes' AND hot_topic = 'yes' ORDER BY article_date DESC LIMIT 3";
and see if that helps as well...