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Infinitesimals

Infinitesimals are fundamental in limit theory and play a key role in limit computation.

定义

Definition of an infinitesimal

If $\lim f(x) = 0$ , then $f(x)$ is called an infinitesimal.

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

$\varepsilon$ (epsilon): Greek letter, read “epsilon”; represents an arbitrarily small positive number.
$\delta$ (delta): Greek letter, read “delta”; represents a positive number depending on $\varepsilon$ .

Examples：

As $x \to 0$ , $x$ , $x^2$ , and $\sin x$ are infinitesimal.
As $n \to \infty$ , $\frac{1}{n}$ and $\frac{1}{n^2}$ are infinitesimal.

性质

A finite sum of infinitesimals is still infinitesimal.
A finite product of infinitesimals is still infinitesimal.
The product of a bounded function and an infinitesimal is infinitesimal.

无穷小的比较

定义

Comparing infinitesimals

Let $\lim \alpha(x) = 0$ , $\lim \beta(x) = 0$ , and $\beta(x) \neq 0$ . Then:

Higher-order infinitesimal: $\lim \frac{\alpha}{\beta} = 0$ , written $\alpha = o(\beta)$ .
Lower-order infinitesimal: $\lim \frac{\alpha}{\beta} = \infty$ , written $\beta = o(\alpha)$ .
Same-order infinitesimal: $\lim \frac{\alpha}{\beta} = C \neq 0$ , written $\alpha = O(\beta)$ .
Equivalent infinitesimal: $\lim \frac{\alpha}{\beta} = 1$ , written $\alpha \sim \beta$ .

$\alpha$ (alpha): Greek letter, read “alpha”; denotes an infinitesimal here.
$\beta$ (beta): Greek letter, read “beta”; denotes an infinitesimal here.
$\gamma$ (gamma): Greek letter, read “gamma”; denotes an infinitesimal here.
$o(\beta)$ : little-o notation, denotes a higher-order infinitesimal.
$O(\beta)$ : big-O notation, denotes a same-order infinitesimal.
$\sim$ : equivalence symbol, denotes equivalent infinitesimals.

比较的例子

As $x \to 0$ :

$x^2$ is a higher-order infinitesimal than $x$ : $\lim \frac{x^2}{x} = 0$ .
$x$ is a lower-order infinitesimal than $x^2$ : $\lim \frac{x}{x^2} = \infty$ .
$2x$ and $x$ are same-order infinitesimals: $\lim \frac{2x}{x} = 2$ .
$\sin x$ and $x$ are equivalent infinitesimals: $\lim \frac{\sin x}{x} = 1$ .

等价无穷小

定义

Equivalent infinitesimals

If $\lim \frac{\alpha}{\beta} = 1$ , then $\alpha$ and $\beta$ are equivalent infinitesimals, written $\alpha \sim \beta$ .

重要性质

Transitivity: If $\alpha \sim \beta$ and $\beta \sim \gamma$ , then $\alpha \sim \gamma$ .
Symmetry: If $\alpha \sim \beta$ , then $\beta \sim \alpha$ .
Substitution: Equivalent infinitesimals can replace each other when computing limits.

常用等价无穷小

When $x \to 0$ :

$\sin x \sim x$
$\tan x \sim x$
$\arcsin x \sim x$
$\arctan x \sim x$
$1 - \cos x \sim \frac{x^2}{2}$
$e^x - 1 \sim x$
$\ln(1 + x) \sim x$
$(1 + x)^\alpha - 1 \sim \alpha x$

等价无穷小的应用

Substitution rule：When computing limits, replace complicated infinitesimals with simpler equivalent ones.

Examples：

$\lim_{x \to 0} \frac{\sin 3x}{x} = \lim_{x \to 0} \frac{3x}{x} = 3$
$\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \lim_{x \to 0} \frac{\frac{x^2}{2}}{x^2} = \frac{1}{2}$

无穷小的阶

定义

Order of an infinitesimal

If $\alpha \sim x^n$ , we say $\alpha$ is an $n$ th-order infinitesimal.

例子

When $x \to 0$ :

$x$ is a first-order infinitesimal.
$x^2$ is a second-order infinitesimal.
$1 - \cos x$ is a second-order infinitesimal (since $1 - \cos x \sim \frac{x^2}{2}$ ).

练习题

练习 1

Determine the relationship between $x^3$ and $x^2$ as $x \to 0$ .

Reference Answer (1 个标签)

infinitesimal

Idea：Evaluate $\lim_{x \to 0} \frac{x^3}{x^2}$ .

Steps：

$\lim_{x \to 0} \frac{x^3}{x^2} = \lim_{x \to 0} x = 0$
Because the limit is $0$ , $x^3$ is a higher-order infinitesimal than $x^2$ .

Answer： $x^3$ is higher order than $x^2$ .

练习 2

Use equivalent infinitesimals to find $\lim_{x \to 0} \frac{\tan x - \sin x}{x^3}$ .

Reference Answer (1 个标签)

infinitesimal

Idea：Apply equivalent infinitesimal substitutions.

Steps：

As $x \to 0$ , $\tan x \sim x$ and $\sin x \sim x$ .
But $\tan x - \sin x$ needs extra handling.
$\tan x - \sin x = \frac{\sin x}{\cos x} - \sin x = \sin x \left(\frac{1}{\cos x} - 1\right) = \sin x \cdot \frac{1 - \cos x}{\cos x}$
As $x \to 0$ , $\sin x \sim x$ , $1 - \cos x \sim \frac{x^2}{2}$ , and $\cos x \to 1$ .
Thus $\tan x - \sin x \sim x \cdot \frac{x^2}{2} = \frac{x^3}{2}$
Therefore $\lim_{x \to 0} \frac{\tan x - \sin x}{x^3} = \lim_{x \to 0} \frac{\frac{x^3}{2}}{x^3} = \frac{1}{2}$

Answer：The limit equals $\frac{1}{2}$ .

练习 3

Decide whether the sequence $x_n = \frac{1}{n^2}$ is an infinitesimal sequence.

Reference Answer (1 个标签)

infinitesimal

Idea：Check whether $\lim_{n \to \infty} \frac{1}{n^2} = 0$ .

Steps：

$\lim_{n \to \infty} \frac{1}{n^2} = 0$
Hence $x_n = \frac{1}{n^2}$ is an infinitesimal sequence.

Answer：It is an infinitesimal sequence.

总结

本文出现的符号

符号	类型	读音/说明	在本文中的含义
$\varepsilon$	希腊字母	Epsilon（伊普西隆）	Arbitrarily small positive number
$\delta$	希腊字母	Delta（德尔塔）	Positive number depending on $\varepsilon$
$\alpha$	希腊字母	Alpha（阿尔法）	Denotes an infinitesimal
$\beta$	希腊字母	Beta（贝塔）	Denotes an infinitesimal
$\gamma$	希腊字母	Gamma（伽马）	Denotes an infinitesimal
$o(\beta)$	数学符号	Little-o notation	Denotes a higher-order infinitesimal
$O(\beta)$	数学符号	Big-O notation	Denotes a same-order infinitesimal
$\sim$	数学符号	Equivalence symbol	Denotes equivalent infinitesimals
$\lim$	数学符号	Limit	Denotes the limit of a function or sequence
$\to$	数学符号	Tends to	Indicates approaching a value

中英对照

中文术语	英文术语	音标	说明
无穷小	infinitesimal	/ˌɪnfɪnɪˈtesɪməl/	极限为 0 的函数或数列
高阶无穷小	higher order infinitesimal	/ˈhaɪə ˈɔːdə ˌɪnfɪnɪˈtesɪməl/	比另一个无穷小更快趋于 0
低阶无穷小	lower order infinitesimal	/ˈləʊə ˈɔːdə ˌɪnfɪnɪˈtesɪməl/	比另一个无穷小更慢趋于 0
同阶无穷小	same order infinitesimal	/seɪm ˈɔːdə ˌɪnfɪnɪˈtesɪməl/	两个无穷小的比值趋于非零常数
等价无穷小	equivalent infinitesimal	/ɪˈkwɪvələnt ˌɪnfɪnɪˈtesɪməl/	两个无穷小的比值趋于 1
无穷小的阶	order of infinitesimal	/ˈɔːdə əv ˌɪnfɪnɪˈtesɪməl/	无穷小与 $x^n$ 的等价关系
小 o 记号	little-o notation	/ˈlɪtəl əʊ nəʊˈteɪʃən/	表示高阶无穷小的记号
大 O 记号	big-O notation	/bɪɡ əʊ nəʊˈteɪʃən/	表示同阶无穷小的记号

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