It can be difficult to lose a friend, especially when with no reason. Sometimes friends will separate other friends because of their own reasons, like they are insecure or want to be popular, etc. and you should not take it personally because it is an issue within them, it does not mean you’re a bad person or have been a bad friend.
You need to ask your friend for an appointment. If she or he agrees to give you time, tell the person that neither of you played a role in your separation and that you want back your friendship. Explain why she or he is an important friend to you. Ask if they see any challenge in this. During your conversation, you may want to avoid criticizing the person who led to your separation, because you don’t know how strong their friendship is. Just focus on your own business, leave alone the other person because you may lose again as a result of speaking badly of them.
But be prepared that you could expect a negative answer as your friend might not yet be ready to be reunited. If that is the case, while it may hurt, if you have been able to say what’s on your mind that is what is important, you did your part to try and repair things. You might find that by sowing the seed, your friendship can flourish again at a later date so consider showing her or him that there is no bad feeling remaining in you. If they accept the friendship offer then you may want to take things slowly to begin with, while both of you build up trust in one another again. I wish you good luck.