Practice a College Board-style free response question on evaluating limits, continuity, and the Intermediate Value Theorem. Write your response, then reveal the model answer to see exactly what earns each point.
Free Response Question · Unit 1 · Limits & Continuity
Let f be the function defined by
f(x) = (x² − 9)/(x − 3) for x ≠ 3, and f(3) = 4.
The function g is continuous on the closed interval [0, 5] with g(0) = −1 and g(5) = 7.
A
Evaluate lim(x→3) f(x). Show the algebraic work that supports your answer.
✓ Model answer
Direct substitution gives 0/0, so factor the numerator: (x² − 9)/(x − 3) = (x − 3)(x + 3)/(x − 3) = x + 3 for x ≠ 3. Therefore lim(x→3) f(x) = 3 + 3 = 6.
Why it scores: Recognizes the indeterminate 0/0 form, factors and cancels correctly, and evaluates the simplified limit to 6. Simply plugging in x = 3 (getting 0/0) without simplifying would not earn credit.
B
Is f continuous at x = 3? Justify your answer using the definition of continuity, and state how f could be redefined to make it continuous there.
✓ Model answer
f is not continuous at x = 3. The definition of continuity requires lim(x→3) f(x) = f(3). Here f(3) = 4 is defined and the limit exists and equals 6, but 6 ≠ 4, so the third condition fails. This is a removable discontinuity; redefining f(3) = 6 (the value of the limit) would make f continuous at x = 3.
Why it scores: States "not continuous," justifies it by comparing the limit (6) to f(3) = 4 against the definition, identifies it as removable, and gives the correct redefinition. A yes/no answer without the definition-based justification would lose credit.
C
Using the given information about g, explain why there must be a value c in the interval [0, 5] such that g(c) = 2.
✓ Model answer
g is continuous on the closed interval [0, 5], and the value 2 lies between g(0) = −1 and g(5) = 7. By the Intermediate Value Theorem, there must exist at least one value c in [0, 5] with g(c) = 2.
Why it scores: Explicitly checks the two IVT hypotheses — continuity on the closed interval AND that 2 is between g(0) and g(5) — before invoking the theorem by name. Omitting either hypothesis, or not naming the IVT, would lose credit.
How to score points on AP Calculus FRQs
Show the algebra. For a 0/0 limit, demonstrate the factoring or rationalizing, then evaluate — don't just state the answer.
Justify continuity with the definition. Compare the limit to the function value against the three conditions.
Name theorems and verify their hypotheses. For the IVT, confirm continuity on the closed interval and that the target value is between the endpoints.
Use correct notation. Write limits with proper lim notation and be precise about one-sided limits.
Answer the question asked. "Justify," "explain," and "show work" each require supporting reasoning, not just a final value.