Is TOT limit computable?

One autumn evening, a boy came to visit the old man. As the boy approached, the old man noticed that the boy was holding one arm behind his back.

"I have a bird," said the boy. "Tell me if it's dead or alive."

The boy knew how to outsmart the old man. The bird in his hand was still alive. But if the old man replied "alive," he would crush the bird's neck. On the other hand, if the old man thought the bird was dead, the boy would open his hand and let the bird fly off.

Replied the old man, "As you wish, my son. As you wish."