This would be an awesome feature for browsers to be able to offer.
Really the only way to know if the user still has the browser window open is to set an interval and ping the server ever so often, when the pinging stops do what ever it is you need to do on the server side.
If you do not need to do something on the server when the client closes the browser, then there really is no real reason to be notified. It would be nice, but not something that is really necessary.