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Infinity

Infinity is a key concept in limit theory and often appears in limit computation.

Definition

If $\lim f(x) = \infty$ , then $f(x)$ is called infinite.

Mathematical wording: For any $M > 0$ , there exists a $\delta > 0$ such that when $0 < \vert x - x_0 \vert < \delta$ , we have $\vert f(x) \vert > M$ .

$M$ : denotes an arbitrarily large positive number used in the definition of infinity.
$\delta$ (delta): Greek letter, read “delta”; denotes a positive number depending on $M$ .

Examples:

As $x \to 0$ , $\frac{1}{x}$ is infinite.
As $x \to \infty$ , $x$ , $x^2$ , and $e^x$ are infinite.

Classification of Infinities

Positive infinity: $\lim f(x) = +\infty$ .
Negative infinity: $\lim f(x) = -\infty$ .
Infinity (unsigned): $\lim f(x) = \infty$ (includes both $+\infty$ and $-\infty$ ).

Relationship between Infinitesimal and Infinity

Basic Relationship

If $\lim f(x) = \infty$ , then $\lim \frac{1}{f(x)} = 0$

If $\lim f(x) = 0$ and $f(x) \neq 0$ , then $\lim \frac{1}{f(x)} = \infty$

In the same process, if $f(x)$ is infinite, then $\frac{1}{f(x)}$ is infinitesimal; conversely, if $f(x)$ is infinitesimal and $f(x) \neq 0$ , then $\frac{1}{f(x)}$ is infinite.

Examples

As $x \to 0$ , $x$ is infinitesimal and $\frac{1}{x}$ is infinite.
As $x \to \infty$ , $\frac{1}{x}$ is infinitesimal and $x$ is infinite.

Practice Problems

Exercise 1

Decide whether $x^2$ is infinite when $x \to \infty$ .

Reference Answer

infinity

Approach:Check $\lim_{x \to \infty} x^2$ .

Steps:

For any $M > 0$ , choose $\delta = \sqrt{M}$ .
When $x > \delta$ , $x^2 > M$ .
Therefore $\lim_{x \to \infty} x^2 = \infty$ .

Answer: $x^2$ is infinite.

Exercise 2

When $x \to 0^+$ , is $\frac{1}{x^2}$ positive infinity or negative infinity?

Reference Answer

infinity

Approach:Analyze the sign and the limit.

Steps:

As $x \to 0^+$ , $x > 0$ , so $x^2 > 0$ .
Hence $\frac{1}{x^2} > 0$ .
And $\lim_{x \to 0^+} \frac{1}{x^2} = +\infty$ .

Answer: $\frac{1}{x^2}$ tends to $+\infty$ .

Exercise 3

Use the relationship between infinitesimals and infinities to judge whether $\frac{1}{x}$ is infinitesimal or infinite as $x \to \infty$ .

Reference Answer

infinity

Approach:Apply the relationship theorem.

Steps:

As $x \to \infty$ , $x$ is infinite.
By the theorem, $\frac{1}{x}$ is infinitesimal.
That is, $\lim_{x \to \infty} \frac{1}{x} = 0$ .

Answer: $\frac{1}{x}$ is infinitesimal.

Summary

Symbols Used in This Article

Symbol notation

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$M$	Mathematical symbol	Arbitrary large positive number	Used to define infinity
$\delta$	Greek letter	Delta (/ˈdɛltə/)	Positive number depending on $M$
$\lim$	Mathematical symbol	Limit	Denotes the limit of a function or sequence
$\to$	Mathematical symbol	Tends to	Indicates approaching a value
$\infty$	Mathematical symbol	Infinity	Represents infinity
$+\infty$	Mathematical symbol	Positive infinity	Represents positive infinity
$-\infty$	Mathematical symbol	Negative infinity	Represents negative infinity

Comprehensive Exercises on Important Limits

Methods for Evaluating Limits

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Infinity

Definition

Classification of Infinities

Relationship between Infinitesimal and Infinity

Basic Relationship

Examples

Practice Problems

Exercise 1

Exercise 2

Exercise 3

Summary

Symbols Used in This Article

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