I had a Golden retriever named Chrissy and had to do that very thing many years ago. She was a great dog and even though hurting and in obvious pain, she was wagging her tail when she could no longer get up on her feet as I approached.It's as if she was telling me, I'm ready to go Dad.
Memories are the greatest gift we have from God. You will always see Chrissy in your mind and think of all the great times you had together.

I am sorry for your loss.