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AP Calculus AB: Limits, Continuity and Derivative Rules — quiz

15 multiple-choice questions covering Limits: one-sided limits, Limit laws, Squeeze Theorem, Horizontal asymptotes, Vertical asymptotes. One answer each; you get a score at the end and the explanation for every question. No account needed.

Updated 2026-11-03. Keys 1–4 pick an answer, Enter moves on. Redo it any time, in order or shuffled.

The questions

Answers and explanations are shown after you finish.

1. Limits: one-sided limits

   A function f satisfies lim_{x→2⁻} f(x) = 5 and lim_{x→2⁺} f(x) = 5, but f(2) is undefined. What can you conclude about lim_{x→2} f(x)?

   • It equals f(2), which must also be 5
   • It equals 5, since both one-sided limits agree (f(2) is irrelevant)
   • It does not exist because f(2) is undefined
   • It cannot be determined without knowing f(2)
2. Limit laws

   Given lim_{x→3} f(x) = 4 and lim_{x→3} g(x) = -2, what is lim_{x→3} [2f(x) - 3g(x)]?

   • 2
   • 14
   • 20
   • -4
3. Squeeze Theorem

   Suppose 1 - x^2 ≤ h(x) ≤ 1 + x^2 for all x near 0. What does the Squeeze Theorem conclude about lim_{x→0} h(x)?

   • 0
   • The limit does not exist
   • 1
   • It cannot be determined without knowing h(x) explicitly
4. Horizontal asymptotes

   Which horizontal asymptote, if any, does f(x) = (4x^3 - x)/(2x^3 + 5) have?

   • No horizontal asymptote
   • y = 2
   • y = 4/5
   • y = 0
5. Vertical asymptotes

   For f(x) = (x+3)/(x^2 - 9), which zero of the denominator does NOT produce a vertical asymptote?

   • x = 3
   • x = -3
   • x = 0
   • x = 9
6. Limits at infinity

   Evaluate lim_{x→∞} (5x^4 - 3x)/(2x^4 + 7x^2).

   • ∞
   • 0
   • 7/2
   • 5/2
7. Discontinuities

   A piecewise function has f(x) = x^2 for x<1 and f(x) = x+3 for x≥1. As x→1⁻, f→1, and as x→1⁺, f→4. What type of discontinuity occurs at x=1?

   • Infinite discontinuity
   • Removable discontinuity
   • No discontinuity; f is continuous at x=1
   • Jump discontinuity
8. Intermediate Value Theorem

   A continuous function f satisfies f(0) = -3 and f(4) = 5. Which conclusion is guaranteed by the Intermediate Value Theorem?

   • f has exactly one zero in (0,4)
   • There exists some c in (0,4) with f(c) = 0
   • f(2) = 1
   • f is differentiable on (0,4)
9. Definition of the derivative

   If f'(5) = -2, what does this tell you about the graph of y = f(x) at x = 5?

   • The function is decreasing on its entire domain
   • f(5) = -2
   • The function has a local maximum at x=5
   • The tangent line to the graph at x=5 has slope -2
10. Derivative rules (product/chain)

    What is the derivative of f(x) = x^3 cos(2x)?

    • -3x^2 sin(2x)
    • 3x^2 cos(2x) - 2x^3 sin(2x)
    • 3x^2 cos(2x) + 2x^3 sin(2x)
    • 3x^2 cos(2x) - x^3 sin(2x)
11. Logarithmic derivatives

    What is d/dx [ln(cos x)]?

    • -sec x
    • 1/cos x
    • -tan x
    • tan x
12. Implicit differentiation

    Find dy/dx for the curve x·y^2 = 4 using implicit differentiation.

    • y/(2x)
    • -2y/x
    • -y/(2x)
    • -y^2/(2x)
13. Differentiability vs. continuity

    Which function is continuous at x=0 but NOT differentiable there because of a vertical tangent (not a corner)?

    • f(x) = 1/x
    • f(x) = x^(1/3)
    • f(x) = |x|
    • f(x) = x^2
14. Mean Value Theorem

    A function f is continuous on [1,5] and differentiable on (1,5), with f(1)=2 and f(5)=14. By the Mean Value Theorem, what is guaranteed about some c in (1,5)?

    • f'(c) = 12 for some c in (1,5)
    • f(c) = 3 for some c in (1,5)
    • f'(c) = 0 for some c in (1,5)
    • f'(c) = 3 for some c in (1,5)
15. L'Hôpital's Rule

    Evaluate lim_{x→0} (e^{2x} - 1)/x using L'Hôpital's Rule.

    • 2
    • 0
    • 1
    • ∞