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Quizzes > Mathematics

Calculus Limits Quiz: Practice Limits and Continuity

Quick, free limits and continuity quiz. Instant results.

Editorial: Review CompletedCreated By: Omar RodriguezUpdated Aug 24, 2025
Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art design of layered graphs and calculus symbols on dark blue background promoting free limits and continuity quiz

This calculus limits quiz helps you practice limits and continuity and see where you need review. Work through quick questions on one-sided limits, limits at infinity, and continuity at a point, with instant feedback. For extra practice, try our limits and continuity quiz, a focused derivative quiz, and an integral quiz as you build skills for tougher problems.

Easy
What is lim??3 (2x + 1)?
7
6
8
5
You can directly substitute x = 3 into 2x + 1 to get 7. This is an example of a limit where the function is continuous at the point. For continuous polynomials, lim??a f(x) = f(a).
Evaluate lim??2 (x² - 4).
4
- 4
2
0
Since x² - 4 is a polynomial, it's continuous everywhere. Substituting x = 2 gives 2² - 4 = 0. Continuous functions allow direct substitution to find limits.
What is lim??5 10?
10
5
Undefined
0
A constant function has the same value at every point, so its limit is that constant. Therefore, lim??5 10 = 10. Constants are trivially continuous.
Find lim??0 (sin x) / x.
1
?
- 1
0
This is a standard trigonometric limit which equals 1. It can be shown using the squeeze theorem or series expansion. This limit is fundamental in calculus.
Medium
Compute lim??2 (x² - 4)/(x - 2).
4
Undefined
0
2
Factor the numerator: (x² - 4) = (x - 2)(x+2). Cancel (x - 2) to get x+2, then substitute x=2, yielding 4. This removes the removable discontinuity.
Evaluate lim??3 (?(x+1) - 2)/(x - 3).
2
1/4
1/2
Undefined
Multiply by the conjugate: (?(x+1)+2)/(?(x+1)+2). Simplifying gives 1/[?(4)+2] = 1/4. This technique removes the 0/0 form.
What is lim??0 (1 - cos x)/(x²)?
1
1/2
2
0
Use the Taylor expansion cos x ? 1 - x²/2 for small x, so the expression ? (x²/2)/x² = 1/2. Alternatively, apply L'Hôpital's rule twice.
Determine lim??? (3x² + 2x)/(x² - 1).
2
3
0
?
Divide numerator and denominator by x²: (3 + 2/x)/(1 - 1/x²). As x??, 2/x and 1/x² ? 0, leaving 3. This is the ratio of leading coefficients.
Hard
Is the function f(x) = { x² if x<2; 4 if x=2; 2x if x>2 } continuous at x=2?
No, because the left limit differs.
No, because the right limit differs.
Yes, because both one-sided limits equal the function value.
No, because the function is not defined at 2.
Left-hand limit: lim??2? x² = 4. Right-hand limit: lim??2? 2x = 4. f(2) = 4, so the function is continuous at 2. Matching one-sided limits and value implies continuity.
Find lim??2? (x - 2)/|x - 2|.
1
- 1
Does not exist
0
For x<2, x - 2 is negative and |x - 2| = - (x - 2), so (x - 2)/|x - 2| = (negative)/(positive) times - 1 = 1. This shows the left-hand limit equals 1.
Evaluate lim??0 (e? - 1)/x.
?
0
e
1
The derivative of e? at x=0 is e? = 1, so by definition lim??0 (e? - 1)/x = 1. This limit corresponds to the instantaneous rate of change.
Expert
For f(x) = 5x - 3, given any ?>0, which ? ensures |f(x) - 7| < ? whenever |x - 2| < ??
? = ?/5
? = 5?
? = ?
? = ?/3
We need |5x - 3 - 7| = |5(x - 2)| = 5|x - 2| < ?, so |x - 2| < ?/5. Thus choosing ? = ?/5 works for the definition. This is the formal ?-? proof for a linear function.
0
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Study Outcomes

  1. Understand Fundamental Limit Concepts -

    Learn the essential definitions and theorems behind limits to confidently navigate any limits quiz and build a strong calculus foundation.

  2. Apply Evaluation Techniques -

    Master algebraic and graphical methods for calculating limits, enhancing your performance on a calculus limits quiz and limit practice test alike.

  3. Analyze Continuity Criteria -

    Examine the formal definition of continuity and identify points of discontinuity to excel in both limits and continuity questions and your continuity quiz.

  4. Solve Real-World Problems -

    Use real-life scenarios to apply limit principles, strengthening problem-solving skills and confidence during your free limits quiz session.

  5. Interpret Boundary Behavior -

    Evaluate one-sided and infinite limits to understand function behavior near boundaries, a critical skill for advanced calculus limits quiz challenges.

  6. Assess Quiz Readiness -

    Review your results and pinpoint areas for improvement, ensuring you're fully prepared for any limit practice test or limits quiz mastery opportunity.

Cheat Sheet

  1. Formal Epsilon-Delta Definition -

    The epsilon-delta definition provides a rigorous framework: limₓ→a f(x)=L means for every ε>0 there's a δ>0 such that |f(x)−L|<ε whenever 0<|x−a|<δ (MIT OpenCourseWare). Use the mnemonic "Epsilons Everywhere, Deltas Determine Distance" to recall roles. Mastering this makes any limits quiz questions feel like a breeze.

  2. One-Sided and Infinite Limits -

    One-sided limits (limₓ→a❻ and limₓ→a❺) examine behavior from left or right; infinite limits describe unbounded growth (Stewart's Calculus). Sketching simple graphs helps you visualize asymptotic behavior - think "approach, never touch." This concept often appears in continuity quiz sections to test boundary insights.

  3. Limit Laws and Algebraic Rules -

    The sum, product, quotient, and power laws let you break complex limits into manageable parts (Khan Academy). For example, limₓ→2(x²+3x)=limₓ→2 x² + limₓ→2 3x = 4 + 6 = 10. Memorize "Sally's Pretty Queen Plays" (Sum, Product, Quotient, Power) for speedy recall on your calculus limits quiz.

  4. Techniques: Factoring and Rationalizing -

    When direct substitution yields 0/0, factor polynomials or rationalize conjugates to simplify expressions (University of California, Berkeley). For instance, limₓ→3 (x²−9)/(x−3) = limₓ→3 (x+3)=6 after factoring. Regular practice with a limit practice test solidifies these algebraic moves.

  5. Continuity and the Intermediate Value Theorem -

    A function is continuous at a if limₓ→a f(x)=f(a); polynomials and sine/cosine are continuous everywhere (Paul's Online Math Notes). The Intermediate Value Theorem guarantees f takes every value between f(a) and f(b) - perfect for tackling limits and continuity questions. This principle underpins many real-world modeling problems.

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