Does a limit exist if the left hand side and the right hand side limit are not the same? Especially when there is only a line to the right side of the limit, but not a line to the left side.

No, the limit does not exist if the left-sided limit does not equal to the right-sided limit. Furthermore, if the limit lets say is x - - > 2, and there is a line from the 2 + direction but not a line from 2 - direction, then the left-sided limit doesn't even exist.