Do you have a pic you could post? It could be a fin tear, which would be good to treat with Fish Protector, Garlic Guard and/or Vitachem. 1-2 drops per gallon daily helps heal the damage and protect against infection.
They get them by rubbing up on decorations or plant that have rough edges or if they are getting finrot. Some bettas have natural splits in their tails but they are very rare yet and are sold as "Doubletail" bettas. Usually it is the first sign of serious finrot problems or an injury. The first thing to do is make sure the water is warm enough and clean. Clean water with the proper temperature is a must and the things that Dave mentioned will help a lot. They can all be found at:
I have known of fish that recovered if their water was kept scrupulously clean and warm enough but if the problem is bacterial medication will have to be used and the sooner the better as once it gets to that stage, the problem is hard to treat and cure. Bettas have lovely flowing fins and can be hard to cure with diseases of the fins. I have seen them nearly lose all their fins and take months to regenerate and it takes a lot of time, effort and patience and they may never heal the same as they were but if you are willing to put the work in the job can be done.
Dave has helped a lot of fish recover from fin problems and I have used his method many times but if they do not respond to it fast, I generally give in and go to using Sulfa drugs on them to get a start on the bacterial possibilities. Triple Sulfa or Trisulfa (made by different manufacturers) are both good but vary a lot in cost. For a small tank you need to divide the dose as most of the packets treat 10 gallon tanks but if your water is very hard you may need to double the strength of the meds anyway so a packet could be used in 5 or 6 gallons. But this only applies in areas with extremely hard water.
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