Limit Points Properties Quiz

1) Which best describes a limit point of A?

A point that is an endpoint of A

A point where a sequence diverges

A point where every neighborhood contains a point of A different from it

Any point inside the set

A limit point is a point where every neighborhood contains a point of A other than the point itself. This is the formal definition.

Explanation

A limit point is a point where every neighborhood contains a point of A other than the point itself. This is the formal definition.

2) Which set has 0 as a limit point?

{0}

(0,1)

{1/n}

{1,2,3}

The sequence 1/n gets arbitrarily close to 0, so every neighborhood of 0 contains points of {1/n}.

Explanation

The sequence 1/n gets arbitrarily close to 0, so every neighborhood of 0 contains points of {1/n}.

3) A finite set has how many limit points?

None

Exactly one

Exactly two

Infinitely many

A finite set has no point that can be approached infinitely closely by other points in the set, so it has no limit points.

Explanation

A finite set has no point that can be approached infinitely closely by other points in the set, so it has no limit points.

4) If p is a limit point of A, then:

P must be inside A

P must be outside A

P must be isolated

Every neighborhood of p contains points of A

A limit point must have every neighborhood containing at least one point of A distinct from p.

Explanation

A limit point must have every neighborhood containing at least one point of A distinct from p.

5) Which set has no limit points?

ℝ

ℤ

(0,1)

[0,1]

The integers ℤ are isolated — every integer has a small neighborhood containing no other integers, so ℤ has no limit points.

Explanation

The integers ℤ are isolated — every integer has a small neighborhood containing no other integers, so ℤ has no limit points.

6) Which can be a boundary point but not a limit point?

A point in (0,1)

A point in {0}

A point in ℤ

A point in (1,2)

An integer is an isolated point. It may lie on the boundary of a set but cannot be a limit point because no other points of the set accumulate near it.

Explanation

An integer is an isolated point. It may lie on the boundary of a set but cannot be a limit point because no other points of the set accumulate near it.

7) Which set has infinitely many limit points?

{1,2,3}

ℤ

(0,1)

{0}

The interval (0,1) has every point inside it as a limit point because points cluster everywhere in the interval.

Explanation

The interval (0,1) has every point inside it as a limit point because points cluster everywhere in the interval.

8) If A ⊆ B, then every limit point of A is also a limit point of B.

True

False

If every neighborhood of x contains points of A, then it also contains points of B since A ⊆ B.

Explanation

If every neighborhood of x contains points of A, then it also contains points of B since A ⊆ B.

9) The point 0 is a limit point of {(-1)^n + 1/n}.

True

False

The sequence oscillates near 1 and −1, not near 0. No neighborhood of 0 contains points from this set.

Explanation

The sequence oscillates near 1 and −1, not near 0. No neighborhood of 0 contains points from this set.

10) The limit points of (0,1] ∪ {2} include 2.

True

False

2 is isolated here — no points of the set get arbitrarily close to 2 except 2 itself, which does not qualify.

Explanation

2 is isolated here — no points of the set get arbitrarily close to 2 except 2 itself, which does not qualify.

11) The set ℚ ∩ (0,1) has every point in [0,1] as a limit point.

True

False

Rational numbers are dense in ℝ, so every point in [0,1] has rational numbers arbitrarily close to it.

Explanation

Rational numbers are dense in ℝ, so every point in [0,1] has rational numbers arbitrarily close to it.

12) A limit point of a sequence must also be the limit of the sequence.

True

False

A sequence may have multiple limit points if it oscillates (e.g., (−1)ⁿ). A limit point does not have to be the sequence's actual limit.

Explanation

A sequence may have multiple limit points if it oscillates (e.g., (−1)ⁿ). A limit point does not have to be the sequence's actual limit.

13) The set of limit points of a set is always closed.

True

False

The set of all limit points (the derived set) is always closed because it contains all its own limit points.

Explanation

The set of all limit points (the derived set) is always closed because it contains all its own limit points.

14) A set can have exactly one limit point.

True

False

Example: {1/n} has exactly one limit point, which is 0.

Explanation

Example: {1/n} has exactly one limit point, which is 0.

15) If a limit point belongs to the interior of a set, every neighborhood contains infinitely many points of the set.

True

False

Interior points have full open intervals around them, and a limit point must have infinitely many points of the set arbitrarily nearby.

Explanation

Interior points have full open intervals around them, and a limit point must have infinitely many points of the set arbitrarily nearby.