They really don't recognize the mirror image because it has no smell. Dog's don't see very well and they have big problems seeing 2D (a mirror is a 2D surface). But a dog should recognize his own leg by the smell of it.
There are dogs that have very limited nerve system, so there are parts of their body that could be cut of without the dog even feeling it. This helps them when fighting - they feel no pain, so they keep on fighting (until they eventually bleed to death). But such breads are generally men-made.
Dogs chase their tail out of fun (like a little child simply running not knowing where or why).
But with this dog there's something really wrong. I'm guessing, because it's from a TV show that this dog was either trained to do that (would be a hell of an actor then) or some strange smell has been sprayed on his leg (a sort of a practical joke by the owner who then filmed it and send it on a home-video show or something).
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