The blower motor on the furnace would not shot off. Even when I turn the thermostat setting to the lowest, the motor keep going. I tried to adjust the settings on the "Fan and Limit Control" switch at no avail. Should I change the thermostat or the limit control switch or is there a trouble with the relay in the motor, and if it is so How can I fix that ?

To test the thermostat and low voltage wiring remove the green wire from the low voltage terminal strip inside the furnace. If the blower stops then it is in the thermostat wiring or the thermostat. IF the blower does not stop then it is in the furnace control system. It could be the a/c fan relay stuck or a bad fan and limit control OR if it is a newer model then the high limit could have tripped BUT since you supply no model numbers this is all the help I can be.