How to cope with friend break ups?

I think my best friend of over 8 years hates me now. She's been ignoring me for weeks after a mutual friend cut me off. I feel like the friend cut me off for valid reasons because I screwed up a few times. At the same time they've also not been the nicest person to me either. Always kinda rubbed me the wrong way. So I think it's okay that we're not friends anymore. (I usually am a good friend, but it's hard to be kind to someone that constantly disrespects you)

The person who's been my best friend for the longest time took the other girl's side. And now I just kind of feel empty. It was easy to dismiss the insults of the other girl, because deep down I thought she was a bitch most of the time. But it's different when it comes from someone who's known you for 8 years. I don't know if she agrees with everything the other girl has been saying about me. It's just been radio silence. I want to ask, but I think the answer will just hurt me even more than just silently drifting apart. I know the two of them have a really strong bond right now so I don't think I can do anything.

How do you move on from a friend break up? Especially one that doesn't have any closure?