The limit at a hole: The limit at a hole is the height of the hole. is undefined, the result would be a hole in the function. Function holes often come about from the impossibility of dividing zero by zero.

Does a limit exist at an open circle?

Nope. The open circle does mean the function is undefined at that particular x-value. However, limits do not care what is actually going on at the value. Limits only care about what happens as we approach it.

Is a limit discontinuous at a hole?

A removable discontinuity exists when the limit of the function exists, but one or both of the other two conditions is not met. The graphical feature that results is often colloquially called a hole.