I have to put my dog down, she's so sick. I don't know how I am going to cope with this.
I have had to do this too with my best friends (My beloved dogs). It is hard. I know how you feel because I have been there more times then I want to remember. 
How do you get through it?
Minute by minute and I remind myself that my best friend is no longer in pain.
That's a good outlook, at least as good as it gets. I'll try to remind myself of that.