Limit Points Quiz

1) Every limit point of a set must be an element of the set

True

False

A limit point may lie outside the set. Example: 0 is a limit point of (0,1).

Explanation

A limit point may lie outside the set. Example: 0 is a limit point of (0,1).

2) A point p is a limit point of set A if every neighborhood of p contains a point of A different from p

True

False

This is the definition of a limit point.

Explanation

This is the definition of a limit point.

3) A limit point must always be a boundary point.

True

False

Interior limit points exist — interior points can be limit points.

Explanation

Interior limit points exist — interior points can be limit points.

4) Isolated points are not limit points

True

False

Isolated points have a neighborhood containing no other points of the set.

Explanation

Isolated points have a neighborhood containing no other points of the set.

5) If a set has a limit point, then it must be infinite.

True

False

A finite set cannot have limit points.

Explanation

A finite set cannot have limit points.

6) The closure of a set contains all its limit points.

True

False

Closure = set + all limit points.

Explanation

Closure = set + all limit points.

7) The limit points of the interval [0,1] include both 0 and 1.

True

False

Every neighborhood of 0 or 1 contains points inside [0,1].

Explanation

Every neighborhood of 0 or 1 contains points inside [0,1].

8) Which of the following sets has 1 as a limit point?

{1}

(1,2)

{1 + 1/n}

{2,3,4}

1 + 1/n → 1, so 1 is a limit point.

Explanation

1 + 1/n → 1, so 1 is a limit point.

9) Which of the following statements is true?

The integers have infinitely many limit points

Any infinite set must have a limit point

Any set with a limit point must be infinite

A limit point must be inside the set

A limit point requires infinitely many nearby points.

Explanation

A limit point requires infinitely many nearby points.

10) If A ⊆ B, then:

Every limit point of A is also a limit point of B

Every limit point of B is a limit point of A

No limit points are shared

Limit points do not relate to subset structure

If every neighborhood contains points from A, it also contains points from B.

Explanation

If every neighborhood contains points from A, it also contains points from B.

11) Which point is a limit point of {(-1)^n + 1/n}?

0

1

-1

Both B and C

The sequence alternates: → 1 (from +1/n), → −1 (from −1 + 1/n). Both are approached.

Explanation

The sequence alternates: → 1 (from +1/n), → −1 (from −1 + 1/n). Both are approached.

12) The set ℚ ∩ (0,1). Which limit points?

Only rationals in (0,1)

Only irrationals in (0,1)

All points in [0,1]

No limit points

Rationals are dense, so every point in [0,1] is a limit point.

Explanation

Rationals are dense, so every point in [0,1] is a limit point.

13) If A has a limit point, then its closure:

Cannot be finite

Must be infinite

Must contain that limit point

Must be open

Closure contains all limit points.

Explanation

Closure contains all limit points.

14) Which of the following is true about boundary and limit points?

Every boundary point is a limit point

Every limit point is a boundary point

Boundary points may or may not be limit points

Boundary points cannot be limit points

Some boundary points may be isolated; some are limit points.

Explanation

Some boundary points may be isolated; some are limit points.

15) A point is not a limit point of A when:

A neighborhood contains no points of A

Every neighborhood contains no points of A

It is in the interior

It is a cluster point

If a neighborhood avoids A completely, the point is not a limit point.

Explanation

If a neighborhood avoids A completely, the point is not a limit point.