God has given man charge or dominion over all things - the birds in the air, the fish in the sea and all crawling and walking things.
Some men are going to have to answer the Almighty for the mis-use they rendered to those they were put in charge - and it is no different - if a dog - you grew up with - turned so sick - that he was miserable and you allowed him to live.
To give life or to take life is ultimately God's but He shares that quality in man to take charge of the lesser animals like for food - and when an animals is sick there is no chance of survival - we know what we must do - even though it wrenches our hearts.
Best to you Bill.
Bud