I had a similar experience with Rincewind. One day I noticed about an inch of his tail was turning brown/black and was no longer flexible. A visit to the vet confirmed he had injured it somehow and most likely it would fall off, which it eventually did with little fuss. I was beside myself during that month, but he didn't seem to notice. The vet suggested using neosporin and increasing soak times. If it's only a small part of the tail, the wound will heal inside separate from the tip and when the scales finally separate, it will fall off before the next shed. Keeping neosporin on the outside helps the healing part to be flexible and free of bacteria, long soaks will help soften the dead skin so the dead piece can fall off.
It will be scary and you'll be anxious, but as long as the black section doesn't grow (indicating the whole tail is infected), it should more or less take care of itself within a month.