How do I comfort my pet when they are dying?

It depends on what they're dying from. If they're really in pain, then it's okay to give them painkillers if they need it. If it's clear that they are elderly or suffering from something that's not necessarily physically painful, then cut them some slack. Give them latitude. 

They need you and they'll let you know what they need. There are also differences in the personalities of animals. Some may want more of you, some might want less of you. There's a myth that wild animals go off and die on their own, either to relieve the group of having to go through the pain of watching them die or some other reason. 

The fact of the matter is, there's not much evidence for that, but it's an eye catcher. Having a companion animal is a two-way street. We control their lives and we are their lifeline, so give them what they need. You can just talk softly to them. You can sit down and see if they come to you if they are able. Sit down next to them and see what they do. It's about paying enough attention to acknowledge who they are, as the individual they are.