Foundations Of Epsilon–Delta Proofs Quiz

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

δ = ε

δ = min(1/2, ε/2)

δ = ε/2

δ = ε²

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

Explanation

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

2) A function fails continuity at a if:

Limit exists

F(a) exists

Limₓ→a f(x) ≠ f(a)

Both one-sided limits equal each other

Continuity requires limit = function value.

Explanation

Continuity requires limit = function value.

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

δ = ε

δ = ε/2

δ = ε|x|

δ = 0

||x| − |0|| = |x|

Explanation

||x| − |0|| = |x|

4) If |x−a| < δ implies |f(x)−L| < ε, then the limit exists and equals L.

True

False

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

5) To show limₓ→2 3x = 6, one valid δ is δ = ε/3.

True

False

|3x − 6| = 3|x − 2|

Explanation

|3x − 6| = 3|x − 2|

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

True

False

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

Explanation

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

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

True

False

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

Explanation

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

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

True

False

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

Explanation

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

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

|3x − 3a| < ε, δ = ε/3

|3x − 12| < ε, δ = ε/3

δ = ε

δ = ε + 4

|3x − 12| = 3|x − 4|

Explanation

|3x − 12| = 3|x − 4|

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

Choose any number larger than ε

Control how close f(x) is to f(a)

Control how close x is to a

Make ε = 0

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

Explanation

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

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

δ = ε

δ = ε/4

δ = 2ε

δ = 4ε

|4x − 4| = 4|x − 1|

Explanation

|4x − 4| = 4|x − 1|

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

δ = ε

δ = ε/3

δ = 3ε

δ = ε − 3

|3x − 6| = 3|x − 2|

Explanation

|3x − 6| = 3|x − 2|

13) Which statement is true about ε–δ proofs?

δ must equal ε

δ must depend only on ε

δ must be positive

δ must be extremely small

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

Explanation

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

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

Avoid δ depending on x

Make ε = 0

Prove differentiability

Remove discontinuities

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

Explanation

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

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

δ = ε

δ = min(1, ε/7)

δ = ε|a|

δ = ε/(2a)

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

Explanation

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

Foundations of Epsilon–Delta Proofs Quiz

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

2)

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

2) A function fails continuity at a if:

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

4) If |x−a| < δ implies |f(x)−L| < ε, then the limit exists and equals L.

5) To show limₓ→2 3x = 6, one valid δ is δ = ε/3.

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

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

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

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

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

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

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

13) Which statement is true about ε–δ proofs?

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

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