Infinity

Alternative Definition of an Infinite Set

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A deeper, positive definition of an infinite set states:

A set A is infinite if it has more than ℵ₀ elements

A set A is infinite if and only if it is equivalent to one of its proper subsets

A set A is infinite if it is not equivalent to any finite set

A set A is infinite if it contains all natural numbers

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Is ∞ a Number?

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In standard mathematics, is the symbol \(\infty\) a number?

Yes — it is the largest real number

No — ∞ is a symbol representing unbounded growth, not a number in the usual sense

Yes — it is a complex number with very large magnitude

No — ∞ is only defined in non-standard analysis

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Behaviour of f(n) = 1/n as n → ∞

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The table below shows the values of \(f(n) = 1/n\) as \(n\) grows.

Behavior of f(x) as x Approaches Infinity
x	f(x) = 1/x
10	0.1
100	0.01
1000	0.001
10000	0.0001

What does \(f(n) = 1/n\) approach as \(n\) tends to infinity?

It approaches 1

It approaches 0, but never actually equals 0

It equals 0 when n = ∞

It approaches ∞

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The ε-Condition for 1/n

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For any small positive number \(\varepsilon = 0.001\), we want \(1/n < \varepsilon\). What is the minimum value of \(N\) such that \(n > N\) guarantees this?

N = 10

N = 100

N = 1 000

N = 10 000

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Behaviour of f(n) = n² as n → ∞

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What happens to \(f(n) = n^2\) as \(n\) takes larger and larger values?

It approaches a fixed large number

It approaches 0

It can be made as large as we please — it tends to +∞

It oscillates between positive and negative values

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Notation: Tending to Infinity

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Which of the following is the correct notation for "\(-n^2\) tends to \(-\infty\) as \(n\) tends to \(\infty\)"?

\(-n^2 = -\infty\) as \(n = \infty\)

\(-n^2 \to -\infty\) as \(n \to \infty\)

\(\lim -n^2 = 0\)

\(-n^2 < -\infty\) for large \(n\)

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Infinite vs Very Large Finite

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Which of the following correctly distinguishes infinite from very large finite?

A very large finite number cannot be counted; an infinite set can

Infinite means uncountably large; very large finite means countably large

A very large finite quantity can in principle be fully counted given time; an infinite set cannot be exhausted by counting

There is no real distinction — very large finite numbers are effectively infinite

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Algebra of Infinity — Core Properties

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The table below summarises the accepted properties of \(\infty\) in extended arithmetic.

Limits Approaching Infinity
Function	Behavior as x → ∞
f(x) = 1/x	Approaches 0
f(x) = x²	Approaches ∞
f(x) = -x	Approaches -∞
f(x) = 5	Remains constant

For any \(x \in \mathbb{R}\) and \(x > 0\), what is \(x \cdot (+\infty)\)?

\(-\infty\)

\(0\)

\(+\infty\)

Undefined

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x + (+∞) for x ∈ ℝ

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For any real number \(x \in \mathbb{R}\), what is \(x + (+\infty)\)?

\(0\)

\(-\infty\)

\(+\infty\)

Undefined

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x/(+∞) for x ∈ ℝ

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For any real number \(x \in \mathbb{R}\), what is \(x \div (+\infty)\)?

\(+\infty\)

\(-\infty\)

\(1\)

\(0\)

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What Is x − (−∞)?

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For \(x \in \mathbb{R}\), what is \(x - (-\infty)\)?

\(-\infty\)

\(+\infty\)

\(0\)

\(x\)

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If x < 0, What Is x · (+∞)?

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If \(x < 0\) (a negative real number), what is \(x \cdot (+\infty)\)?

\(+\infty\)

\(-\infty\)

\(0\)

\(x\)

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What Is (+∞)(+∞)?

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What is \((+\infty) \times (+\infty)\)?

\(0\)

\(1\)

\(+\infty\)

Undefined

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Indeterminate Forms Involving ∞

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Expressions involving \(\infty\) that are not defined and cannot be determined without further analysis are called indeterminate forms. Which of the following are indeterminate forms? (Select all that apply.)

\(\infty - \infty\)

\(\infty / \infty\)

\(\infty^0\)

\(x + \infty\) for \(x \in \mathbb{R}\)

\(0 \cdot \infty\)

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Explanation

Match Operations with ∞ to Their Results

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To complete the line match

Click on an item in the first group
Click on an item in the second group
To delete a match, double click on a line

Expression

x + (+∞), x ∈ ℝ

x − (+∞), x ∈ ℝ

x / (+∞), x ∈ ℝ

(+∞)(−∞)

Result

+∞

−∞

0

Match each expression to its result (assuming \(x \in \mathbb{R}\) where applicable).

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How Many Kinds of Infinity Are There?

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Is there only one kind of infinity, or are there multiple distinct infinities?

There is only one kind of infinity, symbolised by ∞

There are exactly two: countable and uncountable

There is an infinite hierarchy of distinct infinities

Infinity is not a mathematical concept — it is only philosophical

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Where Do Real Numbers Lie on the Extended Real Line?

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In the extended real number system, how is the position of any real number \(x\) described?

\(x > +\infty\)

\(-\infty < x < +\infty\), i.e., \(x \in (-\infty, +\infty)\)

\(x = \infty\) for large x

\(x \ge 0\) always

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Sort These Expressions: Defined or Indeterminate?

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Expression

×∞ / ∞

×5 + ∞

×∞ − ∞

×0 × ∞

×3 / ∞

×∞⁰

Defined

Indeterminate

Classify each expression as Defined (has a determinate value) or Indeterminate (cannot be determined from the form alone).

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Precise Meaning of f(n) = 1/n → 0

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The statement "\(f(n) = 1/n\) tends to 0 as \(n \to \infty\)" means precisely:

1/n = 0 for n = ∞

For any positive ε, however small, we can find N such that 1/n < ε whenever n > N

1/n = 0 for all n > 1000

1/n decreases and eventually becomes negative

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Sort the Steps: Proving 1/n → 0 Rigorously

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Please drag and drop the options to sort them

Sort the following steps into the correct logical order to rigorously prove that \(1/n \to 0\).

Conclude that 1/n < ε for all n > N

Choose N ≥ 1/ε (e.g., N = ⌈1/ε⌉)

Let ε > 0 be any arbitrarily small positive number

Observe that n > N ≥ 1/ε implies 1/n < ε

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Alternative Definition of an Infinite Set

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Explanation

Is ∞ a Number?

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Explanation

Behaviour of f(n) = 1/n as n → ∞

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Explanation

The ε-Condition for 1/n

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Behaviour of f(n) = n² as n → ∞

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Explanation

Notation: Tending to Infinity

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Explanation

Infinite vs Very Large Finite

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Algebra of Infinity — Core Properties

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Explanation

x + (+∞) for x ∈ ℝ

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Explanation

x/(+∞) for x ∈ ℝ

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Explanation

What Is x − (−∞)?

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Explanation

If x < 0, What Is x · (+∞)?

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Explanation

What Is (+∞)(+∞)?

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Explanation

Indeterminate Forms Involving ∞

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Explanation

Match Operations with ∞ to Their Results

Expression

Result

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Explanation

How Many Kinds of Infinity Are There?

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Explanation

Where Do Real Numbers Lie on the Extended Real Line?

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Explanation

Sort These Expressions: Defined or Indeterminate?

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Explanation

Precise Meaning of f(n) = 1/n → 0

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Explanation

Sort the Steps: Proving 1/n → 0 Rigorously

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