Unlock hundreds more features
Save your Quiz to the Dashboard
View and Export Results
Use AI to Create Quizzes and Analyse Results

OR
Sign inSign in with Facebook
Sign inSign in with Google
Quizzes > Mathematics

AP Calculus Unit 1 Test: Limits and Continuity Practice

Quick, free AP Calc Unit 1 quiz. Instant results.

Editorial: Review CompletedCreated By: Keyoka AkinsUpdated Aug 25, 2025
Difficulty: Moderate
Grade: Grade 12
Study OutcomesCheat Sheet
Colorful paper art promoting a Calculus AB Kickoff quiz for high school students.

This AP Calculus AB Unit 1 quiz helps you check your understanding of limits, one-sided limits, and continuity. Answer 20 quick questions, get instant results, and see where to focus next. After this, try the AP Calculus unit 2 test or broaden your practice with a calculus quiz; if you want a quick review first, explore pre calculus questions.

Evaluate lim_{x -> 3} (2x^2 - x - 3).
15
12
9
14
undefined
Compute lim_{x -> 2} (x^2 - 4)/(x - 2).
6
2
4
8
undefined
For f(x) = { x + 1 if x < 0, 2x - 1 if x >= 0 }, find lim_{x -> 0^-} f(x).
-2
1
-1
0
undefined
Find lim_{x -> 1} 1/(x - 1)^2.
0
1
Infinity
-Infinity
undefined
Which x-values are excluded from the domain of f(x) = (x^2 - 9)/(x^2 - 5x + 6)?
x = 2 and x = 3
x = -3 and x = 3
x = 3 only
x = -3 and x = 2
undefined
By the Intermediate Value Theorem, which statement is guaranteed? f is continuous on [1,4], f(1) = -2, f(4) = 5.
There exists c in (1,4) with f(c) = 0.
f(c) = 10 for some c in (1,4).
f has a maximum at x = 4.
f is differentiable on (1,4).
undefined
From the graph, as x approaches 2 from the left, f(x) -> 3, and from the right, f(x) -> 5. What is lim_{x->2} f(x)?
4
3
5
Does not exist
undefined
Determine lim_{x -> ∞} (5x^3 - 2x + 7)/(10x^3 + x - 4).
5/10 x = 1/2 x
1/2
0
Infinity
undefined
Evaluate lim_{x -> 0} (√(9 + x) - 3)/x.
DNE
3
1/3
1/6
undefined
Using squeeze theorem, lim_{x -> 0} x^2 sin(1/x).
DNE
1
0
Oscillates between -1 and 1
undefined
Find k so that f(x) = { kx + 1 if x <= 2, x^2 - 1 if x > 2 } is continuous at x = 2.
k = 1.5
k = 2
k = 1
k = 3
undefined
For f(x) = (x^2 - 1)/(x - 1), define f(1) to remove a discontinuity and make f continuous at x = 1.
f(1) = 2
f(1) = undefined
f(1) = 1
f(1) = 0
undefined
Classify the discontinuity at x = 1 for f(x) = { x if x < 1, x + 2 if x >= 1 }.
Jump
Removable
No discontinuity
Infinite
undefined
Compute lim_{x -> 0} (1 - cos x)/x^2.
1
0
1/2
DNE
undefined
Evaluate lim_{x -> 0} sin(1/x).
1
0
Infinity
Does not exist
undefined
Find a and b so f(x) = { ax + b if x <= 1, x^2 + 1 if x > 1 } is continuous and differentiable at x = 1 and f(1) = 2.
a = 2, b = 0
a = 1, b = 1
a = 0, b = 2
a = -1, b = 3
undefined
Evaluate the sign of lim_{x -> 2^-} (x - 2)/(x - 2)^2.
Does not exist (no sign)
-Infinity
Infinity
0
undefined
Evaluate lim_{x -> 0^+} ln(x).
-Infinity
Infinity
0
Does not exist (oscillates)
undefined
If lim_{x->2} f(x) = 4 and lim_{x->2} g(x) = 0 with g(x) != 0 near 2, what is lim_{x->2} f(x)/g(x)?
0
4
Does not exist in general
Infinity or -Infinity (diverges)
undefined
Compute lim_{x -> 0} (x - sin x)/x^3.
1/6
0
DNE
1/3
undefined
0

Study Outcomes

  1. Understand fundamental differentiation concepts and their applications.
  2. Apply differentiation rules to compute derivatives of various functions.
  3. Analyze the behavior of functions using derivative tests.
  4. Evaluate definite integrals to determine areas under curves.
  5. Interpret the relationship between derivatives and integrals in solving calculus problems.

AP Calculus AB Unit 1 Practice Test Cheat Sheet

  1. Master the Concept of Limits - Limits show what value a function gets closer to as its input zeroes in on a point, even if the function never actually reaches it. They form the backbone of continuity and the launch pad into derivatives, so mastering them early is a calculus hack. Dive in here:
  2. Evaluate Limits Numerically, Graphically & Analytically - Tables help you spot patterns, graphs let you visualize behavior, and algebraic techniques sharpen your analytical prowess. Juggling all three boosts your toolkit when one method alone hits a dead end. Level up your skills at
  3. Grasp Continuity and Its Tests - A function is continuous if you can draw it without lifting your pencil; checking limits from both sides and matching the function's defined value ensures smooth behavior. Spotting holes, jumps, and asymptotes makes you a graphing guru. Learn how it all ties together at
  4. Harness the Intermediate Value Theorem - If a function is continuous on [a, b], it must take on every value between f(a) and f(b). It's like guaranteeing you'll taste every flavor at an ice cream buffet - no skips allowed! See why this theorem is root-perfect here:
  5. Explore Infinite Limits & Limits at Infinity - Watch functions skyrocket to ±∞ near vertical asymptotes or settle into horizontal asymptotes out at the horizon. Recognizing these behaviors is key to graphing like a pro and predicting long-term trends. Get the lowdown at
  6. Apply Algebraic Properties to Simplify Limits - Use the sum, difference, product, and quotient rules to break complex limits into bite-sized problems. These shortcuts turn messy expressions into straightforward calculations, saving time and headaches. Watch the magic unfold:
  7. Squeeze (Sandwich) Theorem - When a tricky function hangs out between two friendlier ones with the same limit, it must share their limit too. This squeeze play is especially handy for taming oscillating or trigonometric functions. Get squeezed into success here:
  8. Spot Different Types of Discontinuities - Jump, removable, infinite - each discontinuity tells a story about function behavior. Understanding them helps you predict weird graph twists and turns like a calculus detective. Explore the tales at
  9. Nail One‑Sided Limits - Sometimes you need to peek from the left or just the right before a cliff edge in a graph. One-sided limits help you analyze behavior near sharp turns and gaps with precision. Check out the view from both sides:
  10. Fix Removable Discontinuities - When a function has a hole (like a lost donut), you can often plug the gap by redefining a single point. This makes the function continuous and donut… I mean function whole again. Get the fill-in at
Make a Quiz (FREE) Copy & Edit This Quiz Create a Quiz with AI

Mathematics Quizzes

Powered by: Quiz Maker