Concept preview

What you will be able to do

1. 01Distinguish a removable discontinuity (a hole, where the two-sided limit exists) from non-removable ones (jumps and infinite discontinuities).
2. 02Use factor-and-cancel to evaluate the limit at a hole and find the value that patches the discontinuity.
3. 03Write an extended or redefined function that is continuous by setting $f(c)$ equal to $\lim_{x\to c} f(x)$.

Not all discontinuities are equal. A vertical asymptote tears the graph apart forever, and a jump splits it into two mismatched heights — neither can be fixed by adjusting a single point. But a hole is different: the graph approaches one clear height from both sides, and the only thing wrong is that the point itself is missing or plotted at the wrong height. Because the limit already exists, we can repair the graph with a single well-chosen dot. This lesson is about recognising these removable discontinuities and computing the exact value that removes them.