The high-limit switch (or limit switch) is a temperature-sensing safety control mounted on the furnace plenum. It has two jobs: it tells the blower when to turn on and off around a heat cycle, and — critically — it shuts the burners off if the air around the heat exchanger climbs past a safe temperature. That overheat cutoff protects the heat exchanger from the kind of thermal stress that cracks it.
When a furnace overheats and trips the limit, the most common cause is restricted airflow, not a bad switch. A clogged filter, closed or blocked supply registers, a dirty blower wheel, or sealed-off return vents all trap heat in the cabinet until the switch intervenes. The visible symptoms are a furnace that short-cycles on the high limit, a blower that runs constantly, or intermittent no heat. In a large share of these calls the switch is fine and is correctly reacting to an airflow problem.
The switch is one of the cheapest parts in the furnace — $4 to $25 — and runs $150 to $400 installed once you add the service call and a half-hour of labor. The real value of the visit is finding why it tripped. A technician who replaces a tripping limit switch without checking the filter, blower, and ductwork has treated the alarm and left the fire: the new switch will trip too. If you're told you need a limit switch, ask what was overheating the furnace and whether the airflow was checked first.