Try double not single quotes:
$min=1;
$max=5;
$result = mysql_query("SELECT * FROM news LIMIT $min, $max");
With double quotes, values will be inserted where your variables appear
so the database will receive
SELECT * FROM news LIMIT 1, 5
also you might want to use ORDER BY to get the records, as they will not necessarily be in order as selected above. If you have a column named "date" for example that captures the date of the article:
SELECT * FROM news
ORDER BY date
LIMIT 1, 5