grab a multimeter and figure out what the normal operation of the switch is. it's probably 12 or 24 vdc. leave the multimeter hooked up and watch it while the system is off, operating normally, and experiencing the problem. the resistor is probably a supervision resistor, so there are at least 3 possible outputs from the switch: open, resistor load, closed/short.
you can then decide the best way to diagnose the problem.
once you know how it should be operating, you can, for example, remove the switch and short or open or use the resistor to trick the furnace in to thinking it is still hot and the fan needs to keep running. if the fan doesn't run properly, then you know the problem is not the limit switch.
Your furnace is shutting down on limit. There can be a number of causes for this.
Start by making sure you have a clean filter in the furnace. A dirty or restrictive filter can cause this issue. Alternatively you could also have a dirty or plugged up A/C evaporator coil which is located above your furnace. This can also cause a furnace to cycle off on high limit, however is uncommon unless you have run your furnace without a filter for prolonged periods of time. Also if you have a high efficiency furnace, there is more of a chance to have the secondary heat exchanger plug up rather than the A/C coil.
You can also have a blower motor issue. The motor can be faulty or seizing up causing it to not spin fast enough to move enough air through the furnace. Another cause may be that you have a faulty limit that is opening prematurely.
The major issue that can cause this problem is a heat exchanger issue. If your furnace is cycling off on high limit, and everything checks out, you may have an internal blockage of the heat exchanger. If this is the case either the heat exchanger or the furnace will need to be replaced. A combustion analysis of the furnace is required to prove this is the issue.
If you happen to have a 2 stage high efficiency furnace you could have a bad ventor motor gasket which is allowing air to be drawn in behind the motor. Big cracks in the condensate collector pan can also cause this but is very uncommon. You can also have the a two stage gas valve high fire solenoid stuck open, but this is also very uncommon and can only happen on 2 stage furnaces. Also if you have a bryant, payne, or carrier high efficiency, you could have a cold spot baffle leakage that is allowing air to be drawn in between the primary and secondary heat exchanger however this is also very uncommon.
If you think your blower is not spinning fast enough you may also have a bad blower motor, or if an ecm motor, the control board may be bad. If its a fixed speed motor and your getting power to it but its not spinning fast enough it could possibly be a bad blower motor. If you decide to change out the blower motor, make sure to get the appropriate replacement capacitor to go with it. I actually did have the same issue on an older carrier high efficiency furnace a few weeks back. It ended up being a bad blower motor, however the motor spun freely and had no indications of seizing or hard starting.
Best Answer
The limit switch should show a resistance of very near zero ohms when it is in the closed state. Sometimes your meter may show one or two ohms of resistance due to the meter leads if it has not been calibrated to null out the wire resistance. This would be more noticeable with a digital meter.
Make sure that you test the limit switch with the wiring removed from the switch. Sometimes if you try to measure a switch that is still connected into the circuit you may see an ohms reading that is above zero when the switch contacts are actually open. This reading coming from the attached circuitry. Under such condition you could incorrectly conclude that the switch was faulty by not noting the other state of the switch.