I am a devout Christian. I have euthanized two pets so far. In both cases, I did so BEFORE the animal's life was unbearable.
It is one thing for a person to decide to suffer. It is another to allow an innocent animal to suffer when we have the power to remove their pain.
My cat Snoopy had a kidney issue. So we gave her sub-Q fluids for 6 months and this improved her quality of life but increased how rapidly her kidneys deteriorated. But at least it made her more comfortable for a while longer. Why did we decide to put her to sleep? Because she no longer seemed to have any joy in her life. Her quality of life had become progressively worse and we decided it was not fair to let her suffer. The vet told us it was the right decision - that too many people wait too long.
With my dog, Annie - we talked to our vet and made sure there were no other options. But Annie, a collie, was 14 years old and was declining rapidly. When she would no longer even eat her favorite treats - we knew it was time.
In both cases, we made sure they were comfortable. We paid extra to have the vet come to our home so they could die in peace, surrounded by familiar scents and by their loved ones. I hope when it is my time to go, it is as peaceful.