Foundations of Epsilon–Delta Proofs Quiz

Reviewed by Editorial Team
The ProProfs editorial team is comprised of experienced subject matter experts. They've collectively created over 10,000 quizzes and lessons, serving over 100 million users. Our team includes in-house content moderators and subject matter experts, as well as a global network of rigorously trained contributors. All adhere to our comprehensive editorial guidelines, ensuring the delivery of high-quality content.
Learn about Our Editorial Process
| By Thames
T
Thames
Community Contributor
Quizzes Created: 7288 | Total Attempts: 9,526,234
| Questions: 15 | Updated: Nov 24, 2025
Please wait...
Question 1 / 15
0 %
0/100
Score 0/100
1) If |x−a| < δ implies |f(x)−L| < ε, then the limit exists and equals L.

Explanation

This is exactly the ε–δ definition of the limit. If the condition holds for every ε > 0, the limit exists and equals L.

Submit
Please wait...
About This Quiz
Foundations Of Epsilondelta Proofs Quiz - Quiz

Think you understand the ε–δ definition of limits and continuity? This quiz helps you apply the formal framework behind limits, piecewise continuity, and δ-selection strategies. You’ll work through examples involving linear functions, learn how to bound expressions, and see how δ depends on ε. These questions guide you through the... see morelogic of proving continuity at a point and identifying when functions fail the ε–δ condition. By the end, you’ll have a clearer and more rigorous understanding of formal continuity proofs! see less

2)
You may optionally provide this to label your report, leaderboard, or certificate.
2) To show limₓ→2 3x = 6, one valid δ is δ = ε/3.

Explanation

|3x − 6| = 3|x − 2|

Submit
3) To prove continuity at a, you must always find an explicit δ in terms of ε.

Explanation

Sometimes δ is found implicitly or through known continuity rules. An explicit formula is not always required.

Submit
4) The ε–δ definition can be used to show that piecewise functions may be continuous at the boundary if both sides agree.

Explanation

If both sides give the same limit, the ε–δ method proves continuity at the boundary point.

Submit
5) If for every ε > 0 there exists a δ depending on both ε and a, then f is uniformly continuous.

Explanation

Uniform continuity requires δ to depend only on ε, not on the point a.

Submit
6) To show limₓ→4 (3x) = 12 using ε–δ, you need:

Explanation

|3x − 12| = 3|x − 4|

Submit
7) The purpose of δ in the ε–δ definition is to:

Explanation

δ puts a restriction on input closeness (x near a).

Submit
8) Which δ works for limₓ→1 (4x − 2) = 2?

Explanation

|4x − 4| = 4|x − 1|

Submit
9) If |x − 2| < δ implies |3x − 6| < ε. What δ works?

Explanation

|3x − 6| = 3|x − 2|

Submit
10) Which statement is true about ε–δ proofs?

Explanation

δ must be positive, but it does not have to equal ε or be extremely small.

Submit
11) In ε–δ proofs, bounding expressions like |x + 3| helps to:

Explanation

Bounding removes x-dependence so δ can be expressed only in terms of ε.

Submit
12) To prove limₓ→a (x²) = a², a good δ is:

Explanation

We often choose δ = min(1, ε/(2|a|+1)) or a similar bound.

Submit
13) For limₓ→1 (1/x) = 1, a correct δ is:

Explanation

Bounding x away from 0 ensures |1/x − 1| can be controlled.

Submit
14) A function fails continuity at a if:

Explanation

Continuity requires limit = function value.

Submit
15) For f(x) = |x|, continuity at x = 0 can be shown by:

Explanation

||x| − |0|| = |x|

Submit
View My Results
Cancel
  • All
    All (15)
  • Unanswered
    Unanswered ()
  • Answered
    Answered ()
If |x−a| < δ implies |f(x)−L| < ε,...
To show limₓ→2 3x = 6, one valid δ is δ = ε/3.
To prove continuity at a, you must always find an explicit δ in terms...
The ε–δ definition can be used to show that piecewise functions...
If for every ε > 0 there exists a δ depending on both ε and a,...
To show limₓ→4 (3x) = 12 using ε–δ, you need:
The purpose of δ in the ε–δ definition is to:
Which δ works for limₓ→1 (4x − 2) = 2?
If |x − 2| < δ implies |3x − 6| < ε. What δ works?
Which statement is true about ε–δ proofs?
In ε–δ proofs, bounding expressions like |x + 3| helps to:
To prove limₓ→a (x²) = a², a good δ is:
For limₓ→1 (1/x) = 1, a correct δ is:
A function fails continuity at a if:
For f(x) = |x|, continuity at x = 0 can be shown by:
Alert!

Advertisement