An attained endpoint can be tested by substituting its coordinate into the function, so its resulting value belongs to the function’s output. An open endpoint does not contribute an attained value at that boundary. Its nearby behavior must instead be described with a one-sided limit, which distinguishes a boundary trend from a value actually included in the function.
A one-sided limit describes how a function behaves as its input approaches an endpoint from within the specified interval. This is necessary when direct substitution is impossible because the function is undefined at the boundary. Comparing the limiting behavior with the endpoint’s domain status helps determine whether the boundary represents an actual function value or only an unattained trend.
Domain restrictions determine whether an endpoint is available for substitution and whether its function value can be included in a conclusion. They also identify boundaries where continuity must be tested through appropriate limiting behavior. Consequently, the same algebraic expression can lead to different endpoint interpretations depending on whether the boundary belongs to the function’s domain.
First, identify the interval boundaries and determine whether each endpoint is included in the domain. Substitute an included endpoint into the function and simplify the result. If direct evaluation is undefined or the boundary is open, examine the appropriate one-sided limit instead. Record separately whether the boundary has an attained value or only limiting behavior.
On a closed interval, evaluating the function at the endpoints supplies boundary values that must be considered when identifying absolute maxima and minima. These results can then be compared with the function’s behavior elsewhere on the interval. Omitting an included endpoint could therefore lead to an incomplete or incorrect conclusion about the function’s largest or smallest value.
Endpoint analysis clarifies whether a graph reaches a boundary or merely approaches it, using attained values, open boundaries, and one-sided limits. In optimization, this information shows whether a boundary candidate is actually available. In graph analysis, it also helps connect domain restrictions, continuity, and boundary behavior into a more accurate interpretation of the function.