Yes, they will but are reluctant to do so without good reason. I had to have my 11 1/2 year old boy put to sleep last year and it was hard. 11 years old is old for a dog, some smaller breeds can live 12 to 15 years, but in general bigger breeds are getting old at 11 years. Good plan would be to get a good vet to check her out in the first place, maybe a simple Op to give her a few more years, but the loss of appetite is not a good sign, good luck,