Part 4/5:
To better illustrate this definition, let's examine a hypothetical graph that includes a removable discontinuity. Suppose the graph resembles a circle at the point ( a ) that is filled in, indicating that the function is defined at that point. Conversely, let’s imagine that there are edges or breaks in the graph at the same value of ( x ) approaching ( a ) from different directions.
If ( f(a^+) ) (the limit from the right) does not equal ( f(a^-) ) (the limit from the left), then the limit does not exist at that point. This is a critical insight, especially when dealing with discontinuous functions, as it helps to acknowledge where our limits can be determined or when they break down.