Types of Limits I

Limits describe how a function behaves as the input approaches a specific value.

This helps understand the function’s behavior near points where it is not defined, such as a gap in the graph where a value is missing, or a discontinuous jump where the graph suddenly shifts to another value.

A common example is a piecewise function, which changes suddenly at a specific input value.

This can be visualized using a smart thermostat switch that keeps the heater off when the temperature is above a set level but switches on instantly once the temperature drops below it. On the graph, this appears as an output of 0 above the set temperature and 1 below it.

This sudden change creates different outcomes depending on the direction from which the input approaches, a concept known as a one-sided limit.

When the input approaches the level from the left or negative side, the switch remains off—this is called the left-hand limit.

When the input approaches from the right or positive side, the switch turns on—this is called the right-hand limit.

Since the left-hand and right-hand limits are not equal, the overall limit at that point does not exist.