AP Calculus AB and BC
Removable vs Jump vs Infinite Discontinuity
A removable discontinuity is a hole where the limit exists but the value is missing or wrong. A jump has two different one-sided limits. An infinite discontinuity has at least one infinite one-sided limit. Only the removable kind can be repaired by redefining a single point.
Removable
Use when: Both one-sided limits agree but the function value is missing or different, so a factor cancelled.
Jump
Use when: Both one-sided limits exist and are finite but disagree, typical of a piecewise function whose pieces do not meet.
Infinite
Use when: At least one one-sided limit is infinite, so there is a vertical asymptote.
Side by side
| Removable | Jump | Infinite | |
|---|---|---|---|
| Two-sided limit | Exists | Does not exist | Does not exist |
| Looks like | A hole | A step | An asymptote |
| Can be repaired | Yes | No | No |
| Typical source | A cancelling factor | A piecewise seam | A denominator zero that does not cancel |
The deciding question is always about the two-sided limit. If it exists, the discontinuity is removable no matter what the function value is. If it fails because the sides disagree by a finite amount, it is a jump. If it fails because the function grows without bound, it is infinite.
For a rational function the algebra tells you directly. Factor the numerator and the denominator: a factor that cancels leaves a hole, and a factor that survives in the denominator gives a vertical asymptote.
The mistake
Assuming every zero of the denominator is an asymptote. In the denominator vanishes at 2, but the factor cancels, so the graph has a hole at and no asymptote at all.
Frequently asked questions
Which discontinuities still have a limit?
Only removable ones. A jump and an infinite discontinuity both mean the two-sided limit does not exist.
In the CED: Unit 1: Limits and Continuity