Practice With Epsilon–Delta Proofs Quiz

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

Rewrite |f(x) − f(a)|

Factor out constants

Choose δ = ε / (slope)

Set δ = 0

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

Explanation

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

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

Bound the denominator away from 0

Find δ involving denominator

Guarantee δ depends only on ε

Set ε = 0

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

Explanation

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

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

F continuous at a

G continuous at f(a)

G differentiable

ε for outer function determines ε for inner

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

Explanation

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

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

Factor x³ − a³ = (x − a)(x² + ax + a²)

Assume |x − a| < 1

Bound x² + ax + a²

δ = ε

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

Explanation

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

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

|7x − 7a| = 7|x − a|

δ = ε/7 works

No δ depends on ε

F is linear and therefore continuous

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

Explanation

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

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6) Which statements about δ selection are true?

δ is chosen so that ε-condition holds

δ is allowed to depend on a bound on x

δ must be optimal

δ must be positive

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

Explanation

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

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

For any ε > 0, f(x) gets within ε of 2

F(3) must equal 2

δ depends on ε

|x − 3| < δ ⇒ |f(x) − 2| < ε

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

Explanation

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

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

F(a) exists

Limₓ→ₐ f(x) exists

Limₓ→ₐ f(x) = f(a)

The function is differentiable at a

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

Explanation

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

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

Output closeness

Input closeness

A positive number

A distance between x and a

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

Explanation

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

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

||x| − |a|| ≤ |x − a|

δ = ε

δ = ε/2

Bounding |x| by 1

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

Explanation

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

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

Factor → |x − 4||x + 4|

Restrict |x − 4| < 1 ⇒ 3 < x < 5

|x + 4| < 9

Require |x − 4| < ε/9

Standard quadratic bounding steps.

Explanation

Standard quadratic bounding steps.

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12) Which statements correctly express the ε–δ definition of limit?

For every ε > 0, there exists δ > 0

If 0 < |x − a| < δ, then |f(x) − L| < ε

F(a) must equal L for the limit to exist

δ depends on ε

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

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

How close x must be to a

How close f(x) must be to f(a)

A positive number

0

δ controls input closeness.

Explanation

δ controls input closeness.

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

δ = ε/3

δ = 3ε

δ = ε

δ ≤ ε/3

|3x − 3a| = 3|x − a|

Explanation

|3x − 3a| = 3|x − a|

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

Jump discontinuity

Infinite oscillation

Infinite limit

Hole in graph

All these forms violate ε–δ condition.

Explanation

All these forms violate ε–δ condition.

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Practice with Epsilon–Delta Proofs Quiz

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

2)

What first name or nickname would you like us to use?

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

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

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

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

6) Which statements about δ selection are true?

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

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

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

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

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

12) Which statements correctly express the ε–δ definition of limit?

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

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

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