LIMIT takes 2 arguments - OFFSET, Number of Rows.
In your case, you would want...
LIMIT 0, 5
LIMIT 5, 5
LIMIT 10, 5
If you always wanted to return 5 rows but start at a different index each time.
Based on your question, change the LIMIT clause to
LIMIT ". (($page - 1) * 5) . ", 5"
Check out MySQL Select & Limit details for more info.