How to fix edge of tile that cracked
It would be easier for people to see what's going on if you move the rug to photograph the whole affected area.  It's impossible to tell the extent or location of the crack.  

Do you have a spare tile, or can you buy one?  Do you know the cause of the crack (like something being dropped on it)?  

With better information, we may be able to give you a better answer, but with no other information, I would guess that either the tile wasn't properly supported, or it was installed on a surface that flexes.  Without fixing the cause, there is no real repair that will last.  

Once the support surface is solid, the only repair that will look good will be replacing the tile, which is also the easiest and fastest solution.  

If you can't find a matching tile, an artist may be able to mask the crack by painting filler to match the tile's pattern.  That could be an improvement, but it will still be visible and won't hold up to wear.