Short on time? This page is the recap. For explanations, the graphs and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: lim (x → c⁻) f(x) is the left-hand limit and lim (x → c⁺) f(x) is the right-hand limit.
Recap
- A graph is unbroken at x = c only if f(c) exists, the limit at c exists, and the two are equal. A discontinuity breaks at least one of these.
- The course names three types: removable, jump and due to a vertical asymptote.
- The type depends on the one-sided limits, not just on whether f(c) is defined.
- A zero denominator shows where to look. Factoring shows which type you have.
Key relationships
| Type | Evidence at x = c | Graph |
|---|---|---|
| Removable | Left and right limits finite and equal (= L); f(c) undefined or f(c) ≠ L | Hole at (c, L) |
| Jump | Left and right limits finite but different | Step from one height to another |
| Vertical asymptote | f(x) → ∞ or −∞ on at least one side | Curve shoots up or down beside x = c |
| No discontinuity | Limit exists and equals f(c) | Unbroken |
Rational functions: a factor (x − c) that cancels gives a hole; a factor that remains in the denominator gives a vertical asymptote.
Absolute values: |x − c| = x − c for x > c and −(x − c) for x < c. This often produces a jump.
Assumptions behind the method
- One-sided limits are found from the rule that applies on that side.
- A table of values suggests the type but does not justify it.
- An infinite limit means the limit does not exist as a number.
Mistakes to avoid
- “f(c) is defined, so f is continuous at c.” Check the limit too.
- “Zero denominator, so vertical asymptote.” Factor and cancel first.
- Calling a jump removable. If the one-sided limits differ, no single value can fix the break.
- Checking only one side before naming the type.
- Misreading ≤ and < when finding f(c) for a piecewise function.
Quick self-check
- Classify the discontinuity of (x² − 25)/(x − 5) at x = 5. (Removable: the limit is 10 but the function is undefined at 5.)
- Classify the discontinuities of (x + 1)/(x² − 1). (Removable at x = −1, where the limit is −1/2; vertical asymptote at x = 1.)
- lim (x → 2⁻) f(x) = 2, lim (x → 2⁺) f(x) = 6 and f(2) = 6. Type? (Jump: finite one-sided limits that differ. f(2) matching one side does not help.)
Next: practice questions.