Could this connection and query class be improved?
convention for PHP class names is PascalCase, so your class should be ConnectionToDb.
you are inconsistent with your indentation (2 vs 4 spaces).
select * is discouraged. Don't just select everything if you only need one column, select only that column.
limit is actually a vector for SQL injection. You should also clean table before using it in a query. Even if the data isn't user supplied right now, as this is a class, which is meant for reuse, you can never be sure how it will be used in the future.
Don't just echo unsanitized data from the database (because it might be user supplied), this can result in an XSS vulnerability.
&lt;/br&gt; isn't valid HTML, use &lt;br&gt; or &lt;br /&gt;
I don't see the benefit of having this as a class. I would just remove make_connection and add the connection as a parameter to run_test_query.
run_test_query isn't a very good name.