Death in any form is not pretty or pleasant, not everyone can deal with death or the dying.
Whether its the animal or a human being, at the point of death, you have to put whats best for the animal, or what the persons wishes are, before anything at all that you feel. At this stage you dont matter.
Many people find themselves getting emotional and opinionated about the situation, as it is finalizing. Those need to step away, make it easier on the family, not harder.
There are many ways everyone dies. With all of us hoping for that quiet sleep we never wake up from. Where in actual fact its not that simple. We are blessed by doctors these days who can keep us as comfortable as possible as we cross that threshold to the next world, in some countries they are gifted with the ability to choose when and how they go.
Before that, make the most of everything, find the positive in every single day and moment. Be grateful that you have those around you, that you do. See the many little things that they do for you. Ask everyone how they are, not just the person who is dying. Those closest are just as effected by the journey.
Currently I am on this journey with my old dog, he is almost 17 years old, if we make it that far. My feelings will be put last, when the meds he is on stops working, straight down to the vet, I will worry about me after, I will have those feelings after, as I light a candle to light his way.
Then and only then, will I take the time to grieve and think about myself. What I will never see in the flesh again, but will forever see in my dreams and heart.