Identifying Limit Points Quiz

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1) Which are true about a limit point of A?

Every deleted neighborhood contains a point of A

The point must belong to A

The point may or may not belong to A

A limit point requires infinitely many points arbitrarily near it

A limit point requires that every deleted neighborhood touches A; it may or may not be in A, and it requires infinitely many points arbitrarily near.

Explanation

A limit point requires that every deleted neighborhood touches A; it may or may not be in A, and it requires infinitely many points arbitrarily near.

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About This Quiz

How well can you recognize a limit point just by examining a set? This quiz challenges you to apply the definition of limit points through sequences, deleted neighborhoods, and clustering behavior. You’ll work with bounded sets, discrete sets, infinite sets, and sequences approaching different values. With examples ranging from integers... see moreto convergent sequences and dense sets, you’ll gain strong insight into when points become limit points. By the end, you'll confidently determine limit points and understand how they relate to closure and topological structure!
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2) Which sets have no limit points?

ℤ

(0,1)

A finite set

{π}

Integers and finite sets are isolated; a singleton like {π} also has no limit points.

Explanation

Integers and finite sets are isolated; a singleton like {π} also has no limit points.

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3) Which are limit points of (0,1)?

0.5

0 and 1 are approached by points of (0,1); 0.5 is an interior limit point.

Explanation

0 and 1 are approached by points of (0,1); 0.5 is an interior limit point.

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4) For a point to be a limit point, which must be true?

The point is approached by points in the set

The point is inside the set

The point has no isolated neighborhood

Every deleted neighborhood contains points of the set

A limit point is approached by the set and every deleted neighborhood intersects it.

Explanation

A limit point is approached by the set and every deleted neighborhood intersects it.

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5) Which sets contain at least one limit point?

Any infinite subset of ℝ

Any compact infinite set

Any finite set

Any set containing an interval

Infinite subsets often accumulate; compact infinite sets must have limit points; any set with an interval has infinitely many.

Explanation

Infinite subsets often accumulate; compact infinite sets must have limit points; any set with an interval has infinitely many.

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6) Limit points may be:

Inside the set

Outside the set

Endpoints

Isolated points

Limit points can occur inside, outside, or at endpoints; isolated points cannot be limit points.

Explanation

Limit points can occur inside, outside, or at endpoints; isolated points cannot be limit points.

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7) Which points are limit points of {1 + 1/n} ∪ (2,3)?

All points in (2,3)

1 + 1/n → 1; every point in (2,3) is a limit point since it's an open interval.

Explanation

1 + 1/n → 1; every point in (2,3) is a limit point since it's an open interval.

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8) Closed sets and limit points:

Closed sets contain all their limit points

A set that contains all its limit points is closed

A set with no limit points is always closed

A closed infinite discrete set must have limit points

Closed sets include limit points; containing all limit points makes a set closed; having no limit points automatically satisfies closure.

Explanation

Closed sets include limit points; containing all limit points makes a set closed; having no limit points automatically satisfies closure.

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9) Limit points of ℚ ∩ (0,1):

All points of (0,1)

All points of [0,1]

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.

10) Limit points of (−1)ⁿ:

−1

No limit point

The sequence alternates between 1 and −1 → both are limit points.

Explanation

The sequence alternates between 1 and −1 → both are limit points.

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11) Limit points of A = {1/n} ∪ {2 + 1/n}:

None of the above

1/n → 0; 2 + 1/n → 2.

Explanation

1/n → 0; 2 + 1/n → 2.

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12) Which sets have 2 as a limit point?

{2 + 1/n}

(1,2)

(2,3)

{2}

Only 2 + 1/n approaches 2.

Explanation

Only 2 + 1/n approaches 2.

13) Which sets can be infinite but have no limit points?

ℤ

{n² : n∈ℕ}

ℕ

(0,1)

ℤ, ℕ, and {n²} are discrete; (0,1) has infinitely many limit points.

Explanation

ℤ, ℕ, and {n²} are discrete; (0,1) has infinitely many limit points.

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14) Limit points of any infinite bounded subset of ℝ:

At least one

Infinitely many

Possibly countably many

Possibly uncountably many

By Bolzano–Weierstrass, such a set has at least one limit point and may have countably or uncountably many.

Explanation

By Bolzano–Weierstrass, such a set has at least one limit point and may have countably or uncountably many.

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15) Closure and limit points:

Closure = set + limit points

Closure is always closed

A set equals its closure iff it contains all limit points

A set with no limit points has closure equal to the set

All four statements are properties of closure and limit points.

Explanation

All four statements are properties of closure and limit points.

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