I think it depends on the circumstances of the breakup. Were you the one to break it up, or did they break up with you?
If you broke it up, and they weren’t happy with it, then you probably have a better chance of being received well. You can try to sell the “I made a mistake” excuse. “I didn’t know what I had.” On the other hand, your ex may want to protect themselves from you again. Still, I think the power is on your side, and the ex may welcome another chance.
If they broke up with you, then it will be harder. I suppose you can try to sell the idea that you have changed, and will not bother them in ways you had before. But what if they lost interest in you, or no longer found you attractive, physically, emotionally, intellectually, or any combination of the above? You’ll have a hard time getting them back, except perhaps as an emotional or sexual dishrag—or if they decide they made a mistake, but haven’t reached out because they don’t want to put you in a position where you may get hurt again. Self-respect would seem to dictate that you don’t even try.
Is it fair? Well, supposedly in love, all is fair. It’s your feelings and your life to do with as you see fit. Yes, there is a risk, but then life is always risky. Some people try to avoid risk, but without emotional risk, is it possible to find deep love? Love requires vulnerability, which is always risky.
If you care about them, that’s a good thing. I think I would show my interest, but in interests of fairness, I would not try to sell them the idea. I would just say I was interested in trying again, and then leave it up to them to make a decisions without further information. If they ask “why,” then you can offer reasons. Otherwise, respect their first impulse.
Of course, if you really do love them, it will be hard to do the ethical thing in leaving them as much choice as possible. So you go for it. Part of me thinks that if you even can ask this question, then your feelings aren’t strong enough to try again. You should be very clear on this—that you will not let any obstacles get in the way. You totally love them enough that there is no question but that you should try.