Short on time? This page is the recap. For explanations, 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) means “the limit as x approaches c of f(x)”; c⁻ and c⁺ mean from the left and from the right.
Recap
- You may change only the value f(c), never the values near c.
- If lim (x → c) f(x) exists and equals L, define (or redefine) f(c) = L. The discontinuity is removable.
- If the one-sided limits differ (jump) or are infinite (vertical asymptote), no value of f(c) helps.
- In a piecewise function, each piece is usually continuous on its own interval. Check only the boundaries.
- At a boundary c: substitute c into both expressions and set them equal. The piece with ≤ or ≥ gives f(c).
Key relationships
| Situation | What to do | Result |
|---|---|---|
| 0/0 at c and the limit is L | Find L by algebra (Topic 1.6) | Define f(c) = L |
| f(c) defined but ≠ limit | Find the limit L | Redefine f(c) = L |
| Nonzero/0 after simplifying | Check each side | Asymptote: not removable |
| One-sided limits differ | None for a fixed formula | Jump: not removable |
| One parameter, one boundary | Left value at c = right value at c | One equation |
| Two parameters, two boundaries | One equation per boundary | Solve simultaneously |
Assumptions behind the method
- Each piece must be continuous on its own interval (polynomials, and roots, exponentials and trig functions on their domains).
- A limit uses values near c only, so the value of f(c) never changes the limit.
- The equation may have one solution, two, or none. All three are possible answers.
Mistakes to avoid
- Saying “removable” without giving the value of f(c).
- Trying to remove a jump by picking one side’s value.
- Setting a piece equal to 0 instead of equal to the other piece.
- Substituting the wrong x at a boundary.
- Losing a solution when the condition is a quadratic in k.
- Inventing a value when the parameter cancels and the equation is false.
- Trusting a calculator plot to show a hole.
Quick self-check
- f(x) = (x² − 49)/(x − 7) for x ≠ 7. What value of f(7) makes f continuous? (14)
- f(x) = kx − 1 for x < 2 and f(x) = x² + k for x ≥ 2. Find k. (2k − 1 = 4 + k, so k = 5; both pieces give 9)
- A function has left limit 2 and right limit 5 at x = c. Can a new value of f(c) make it continuous? (No. The two-sided limit does not exist, so no single value works.)
Next: practice questions.