Scope would replace the var completely. If your constructor is working, you are on a PHP5.x environment, so members should be declared as
private $dbhost; for example. Don't use public in PHP for non-final members ever. Since its datatype weak its more work to hack together a __set than it is to provide accessors and mutators for each member.
Constructors do not return results, they return a reference to the created object (which I believe is actually copied from the new to the variable, and void is actually returned from the __construct). You cannot return $this->connect or $this->selectdb, and in a standard method/function you can only return one.
I wouldn't have any of these as members at all. That exposes a privacy problem, since var_dump and print_r can both see private members, this poses a security risk on systems that allow external developers to build on. Just passing the parameters is fine, all you need to track is a $connection variable.
Also, I should mention that you are better off not exiting / dying from a method at all. You would be better throwing an exception and letting the calling scope determine what do to with it.