I've had to do this several times. LJ's suggestion that you replace the corner is right on the mark. I've had success with cutting out the damaged part, carefully cutting replacement to fit well. I then cut a piece of medium to heavy fabric large enough to insert under the good carpeton as many sides as possible. I use contact cement to connect the good carpet to the cloth and to the patch. If you had access to carpet seaming tape and a heater, that would be a better repair. One thing that helps to make the repair blend in is to carefully trim the patch, then if needed, use small sissors to make the edges level. Make sure to clean everything out with a vaccum before you apply glue. Good luck!