There are many reasons to use Ajax on a website as it allows the content to load into the page as needed.  In the past i’ve always made a separate file for handling all these things, and performance wise they may be about the same. However, there is a built in way to do these things the official WordPress way that was actually easier to work with than creating new services.

WordPress Documentation
Demo example

Summarizing, there is a “wp_ajax_[ACTION]” hook that allows server side processing of the ajax request.

add_action( 'wp_ajax_nopriv_[ACTION]', 'ajax_process_function' );
add_action( 'wp_ajax_[ACTION]', 'ajax_process_function' );
function ajax_process_function() {
	$filterpage = $_REQUEST['page'];
	$filterformat = $_REQUEST['format'];
	echo "Page:".$filterpage." Format:".$filterformat;
	die();
}

This is accompanied by a jQuery ajax request that looks like:

$.ajax({
	url: "/wp-admin/admin-ajax.php",
	type: 'post',
	data: {
		action: '[ACTION]',
		filterpage : 1,
		filterformat : "format"
	},
	success: function( html ) {
		if(html){
			$('#container').html( html );
		}else{
			alert("no content");
		}
	}
});