Practice with Epsilon–Delta Proofs Quiz

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| Questions: 15 | Updated: Nov 24, 2025
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1) Which statements correctly express the ε–δ definition of limit?

Explanation

The ε–δ definition requires ε > 0, a corresponding δ > 0, and 0 < |x − a| < δ ⇒ |f(x) − L| < ε; δ depends on ε.

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About This Quiz
Practice With Epsilondelta Proofs Quiz - Quiz

Ready to apply ε–δ reasoning to a variety of functions? This quiz gives you hands-on practice with the limit definition using deleted neighborhoods, bounding techniques, rational functions, absolute values, and composites. You’ll explore how δ is chosen to make output closeness match ε, how limits behave near specific points, and... see morehow discontinuities reveal themselves through ε–δ failures. By the end, you’ll feel confident constructing ε–δ proofs and understanding how they guarantee continuity and limit behavior! see less

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2) Which of the following must be true for f to be continuous at a?

Explanation

Continuity requires existence of f(a), existence of the limit, and equality lim f(x) = f(a).

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3) In the ε–δ definition, ε represents:

Explanation

ε measures how close f(x) must be to L.

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4) In the ε–δ definition, δ represents:

Explanation

δ controls input closeness.

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5) For f(x) = 3x, which δ choices work to show continuity at a?

Explanation

|3x − 3a| = 3|x − a|

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6) Which are valid steps in an ε–δ proof for linear functions?

Explanation

Linear functions yield |m||x − a|; we divide by slope; δ = 0 is invalid.

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7) Which steps are required in proving continuity of a rational function at a point where the denominator ≠ 0?

Explanation

We ensure denominator stays nonzero, control δ, and ε≠0.

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8) For f(x) = |x|, continuity at any a is shown using:

Explanation

Key inequality: ||x| − |a|| ≤ |x − a|.

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9) Which are needed to show continuity of composites g(f(x))?

Explanation

Composite continuity requires f continuous at a, g at f(a), and adjusting ε.

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10) For f(x) = x³, which are necessary strategies for an ε–δ proof?

Explanation

We use factoring, bounding; δ = ε seldom works.

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11) Which situations produce discontinuity detectable by ε–δ?

Explanation

All these forms violate ε–δ condition.

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12) To show f(x) = 7x − 4 is continuous using ε–δ. Which reasoning is correct?

Explanation

Factor to 7|x − a|; δ = ε/7; linear functions are continuous.

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13) For f(x) = x² at a = 4, we want |x² − 16| < ε. Which steps are valid?

Explanation

Standard quadratic bounding steps.

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14) Which statements about δ selection are true?

Explanation

δ must work, may depend on bounds, must be positive.

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15) If limₓ→3 f(x) = 2, what must be true?

Explanation

Limit definition conditions; f(3) need not equal 2.

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Which statements correctly express the ε–δ...
Which of the following must be true for f to be continuous at a?
In the ε–δ definition, ε represents:
In the ε–δ definition, δ represents:
For f(x) = 3x, which δ choices work to show continuity at a?
Which are valid steps in an ε–δ proof for linear functions?
Which steps are required in proving continuity of a rational function...
For f(x) = |x|, continuity at any a is shown using:
Which are needed to show continuity of composites g(f(x))?
For f(x) = x³, which are necessary strategies for an ε–δ proof?
Which situations produce discontinuity detectable by ε–δ?
To show f(x) = 7x − 4 is continuous using ε–δ. Which reasoning...
For f(x) = x² at a = 4, we want |x² − 16| < ε. Which steps are...
Which statements about δ selection are true?
If limₓ→3 f(x) = 2, what must be true?
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