Infinity

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

定义

Definition of infinity

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 $\vert f(x) \vert > M$.

Examples：

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

无穷大的分类

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

无穷小与无穷大的关系

基本关系

Relationship between infinitesimal and infinity

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.

例子

• 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.

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练习题

练习 1

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

参考答案

Idea：Check $\lim_{x \to \infty} x^2$.

Steps：

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

Answer：$x^2$ is infinite.

练习 2

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

参考答案

Idea：Analyze the sign and the limit.

Steps：

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

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

练习 3

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

参考答案

Idea：Apply the relationship theorem.

Steps：

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

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

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总结

本文出现的符号

符号	类型	读音/说明	在本文中的含义
$M$	数学符号	Arbitrary large positive number	Used to define infinity
$\delta$	希腊字母	Delta（德尔塔）	Positive number depending on $M$
$\lim$	数学符号	Limit	Denotes the limit of a function or sequence
$\to$	数学符号	Tends to	Indicates approaching a value
$\infty$	数学符号	Infinity	Represents infinity
$+\infty$	数学符号	Positive infinity	Represents positive infinity
$-\infty$	数学符号	Negative infinity	Represents negative infinity

中英对照

中文术语	英文术语	音标	说明
无穷大	infinity	/ɪnˈfɪnɪti/	极限为无穷大的函数或数列
正无穷大	positive infinity	/ˈpɒzətɪv ɪnˈfɪnɪti/	极限为正无穷大
负无穷大	negative infinity	/ˈneɡətɪv ɪnˈfɪnɪti/	极限为负无穷大
无穷小	infinitesimal	/ˌɪnfɪnɪˈtesɪməl/	极限为 0 的函数或数列

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