All you can do is accept the fact that the animal is going to go soon and to make the best of the time it has left to live. The hardest part of putting down an animal is deciding when it's the right time. On one hand, you don't want it to go before it's suffering makes life not worth living. On the other hand, you have to decide when the animal's life has gotten bad enough to justify ending it's life altogether to stop the suffering along with everything else.
With cats, it can get ugly depending on the demeanor of the cat and how sick it is before it dies. When my twin sister and I were born, my mother already had a cat, and it was put to sleep when I was about 6 or so because it was old and ill. The cat was a sort of one-person cat, it only liked my mom and anyone else was an annoyance, so it wasn't friendly to begin with. But when it got sick, it got very irritable. When my mom brought my aunt over to help her put the cat in a kennel and put it to sleep, the cat was so sick and uneasy that it freaked out and scratched my aunt down inside of her wrist, at least 4 inches long, I don't remember, but the point is that the cat was unhappy and prone to acts like that because of it's poor condition. That's when an animal would be better off put to sleep than living a life of constant discomfort.
Of course, my friend just had his old cat put down because it was very old and arthritic. The cat was very nice and affectionate, even up to the day it died, but it was obvious that she was in really bad condition. Her fur wasn't coming in well, and she couldn't even really jump up onto a couch without being picked up. It just spent all day lying around, even more than cats normally do. What I'm saying is that even if an animal isn't obviously begging for death, it doesn't mean that putting it down is the wrong thing to do.
