Detailed explanation of infinitesimal and infinite

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definition

If the function $f(x)$ 满足 $\underset{\begin{array}{c}x\to x_0\\ (x\to \infty )\end{array}}{\lim }f(x)=0$ , then$f(x)$ 为 $x\to x_0$(orx $x\to$infinitesimalwhen$\infty \; )$

It can be seen from the definition that the essence of infinitesimal is a function. The function value of this function approaches 0 0 infinitely during a certain change of the independent variable.$0$ . Infinitely is not a very small number

The relationship between infinitesimal and limit

$\underset{\begin{array}{c}x\to x_0\\ (x\to \infty )\end{array}}{\lim }f(x)=$The necessary and sufficient conditions for$A$$f\left(x\right)=A+\alpha$ , where$\alpha$ 为 $x\to x_0$(orx $x\to$infinitesimal when$\infty \; )$

This theorem is easily proved by the addition rule of the limit

infinitesimal comparison

To $x\to 0$ hours,$x$ and$x^2$ are infinitesimal, but$\underset{x\to 0}{\lim }\frac{x}{x^2}=\infty$ . Because at$x\to 0$ process,$x^2$ tends to$0$ is faster than$xfast$ _

According to the different situations of the limits of the ratio of two infinitesimals, it reflects that different infinitesimals tend to $0$ speed

Assume $a$ ,$\beta$ is the infinitesimal value in the same change process of the independent variable. According to$\lim \frac{}{b}$The value of

• $0$ : Higher order infinitesimal
  • If there exists $k>0$使$\lim \frac{}{b^k}=c\ne 0$ , then$k$ order infinitesimal
• Not $Constants\; of\; 0$ : infinitesimals of the same order
  • If the constant is $1$ , then it is equivalent to infinitesimal
• $\infty$ : low-order infinitesimal

Higher order infinitesimal

若$\lim \frac{}{b}=0$ is called “$\alpha$ is greater than$"The\; infinitesimal\; of\; the\; higher\; order\; of\; \beta$ ", denoted as$a=o\left(b\right)$

Lower order infinitesimal

若$\lim \frac{}{b}=\infty$ is called “$\alpha$ is greater thanβ \betaThe infinitesimal of the lower order of$\beta \; ”$

If $\alpha$ is greater than$The\; higher\; order\; of\; \beta$ is infinitesimal, then$\beta$ is greater thanα \alphaThe lower order infinitesimal of$\alpha$

infinitesimal of the same order

若$\lim \frac{}{b}=c\ne 0$ is called “$\alpha$与$\beta$ is an infinitesimal of the same order”

k order infinitesimal

若$\lim \frac{}{b^k}=c\ne 0,k>0$ is called “$\alpha$ is about$\beta$的kkInfinitely small of order$k\; ”$

Equivalent to infinitesimal

definition

若$\lim \frac{}{b}=1$ is called “$\alpha$与$\beta$ is equivalent to infinitesimal", denoted as$a\sim b$

nature

• Reflexivity: $a\sim a$
• Symmetry: If $a\sim \beta$则$b\sim a$
• Transitivity: If $a\sim b,b\sim \gamma \in \alpha$ ∼$a\sim c$

Common equivalent infinitesimal

$x\to 0$ o'clock

• $\sin x\sim x$

  proving process

  As shown in the figure, there is an angle $x\in (-\frac{}{2},0)\cup (0,\frac{}{2})$
  

  It can be seen from the figure that when $x\to When\; 0$ , there is always
  $S_{△AOB}<S_{SectorA}<S_{△AOC}$
  $\therefore \frac{1}{2}|\sin x|<\frac{1}{2}|x|<\frac{1}{2}|\tan x|$
  $\therefore |\sin x|<|x|<\frac{|\sin x|}{\cos x},\cos x>0$
  $\therefore \cos x<|\frac{\sin x}{x}|<1$
  $\therefore \cos x<\frac{\sin x}{x}<1,\frac{\sin x}{x}>0$
  $\because \underset{x\to 0}{\lim }\cos x=1$
  Fromthe pinch criterion,$\underset{x\to 0}{\lim }\frac{\sin x}{x}=1$
  $\therefore \sin x\sim x$

• $\tan x\sim x$

  proving process

  $\because$
  $$\begin{array}{rl}\underset{x\to 0}{\lim }\frac{\tan x}{x}& =\underset{x\to 0}{\lim }\frac{\sin x}{x\cos x}\\ & =\underset{x\to 0}{\lim }\frac{\sin x}{x}\cdot \underset{x\to 0}{\lim }\frac{1}{\cos x}\\ & =1\end{array}$$
  $\therefore \tan x\sim x$

• $\arcsin x\sim x$

  proving process

  $\because$
  $$\begin{array}{rl}\underset{x\to 0}{\lim }\frac{\arcsin x}{x}& =\underset{x\to 0}{\lim }\frac{(\arcsin x{)}^{\prime }}{(x{)}^{\prime }}\\ & =\underset{x\to 0}{\lim }\frac{1}{\sqrt{1-x^2}}\\ & =1\end{array}$$
  $\therefore \arcsin x\sim x$

• $a^x-1\sim x\ln a$

  proving process

  $\because$
  $$\begin{array}{rl}\underset{x\to 0}{\lim }\frac{a^x-1}{x\ln a}& =\underset{x\to 0}{\lim }\frac{(a^x-1{)}^{\prime }}{(x\ln a{)}^{\prime }}\\ & =\underset{x\to 0}{\lim }\frac{a^x\ln a}{\ln a}\\ & =1\end{array}$$
  $\therefore a^x-1\sim x\ln a$

  • $e^x-1\sim x$
  • $\ln (1+x)\sim x$

• $1-\cos x\sim \frac{x^2}{2}$

  proving process

  $\because$
  $$\begin{array}{rl}\underset{x\to 0}{\lim }\frac{1-\cos x}{\frac{1}{2}{x}^2}& =\underset{x\to 0}{\lim }\frac{(1-\cos x{)}^{\prime }}{(\frac{1}{2}{x}^2{)}^{\prime }}\\ & =\underset{x\to 0}{\lim }\frac{\sin x}{x}\\ & =1\end{array}$$
  $\therefore 1-\cos x\sim \frac{1}{2}{x}^2$

• $(1+x{)}^a-1\sim ax$

  proving process

  $\because$
  $$\begin{array}{rl}\underset{x\to 0}{\lim }\frac{(1+x{)}^a-1}{ax}& =\underset{x\to 0}{\lim }\frac{[(1+x{)}^a-1{]}^{\prime }}{(ax{)}^{\prime }}\\ & =\underset{x\to 0}{\lim }\frac{a(1+x{)}^{a-1}}{a}\\ & =1\end{array}$$
  $\therefore (1+x{)}^a-1\sim ax$

theorem

Theorem 1: Necessary and sufficient conditions
$a\sim$The necessary and sufficient conditions for$\beta$$b=a+o\left(a\right)$

Theorem 2: Equivalent substitution
If $a\sim b,\tilde{a}\sim \tilde{b}$则$\lim \frac{}{b}=\lim \frac{\tilde{a}}{b}=\lim \frac{}{\tilde{b}}=\lim \frac{\tilde{a}}{\tilde{b}}$

Using the above two theorems is very helpful for finding the limit, such as finding the limit $\underset{x\to 0}{\lim }\frac{2}{x+x^2+x^3}$
Because $x=x+o\left(x\right)$ , where$o(x)=x^2+x^{3So}$
x ∼ x + x 2 + x$x\sim x+x^2+x^3$
所以 $\underset{x\to 0}{\lim }\frac{2}{x+x^2+x^3}=\underset{x\to 0}{\lim }\frac{2}{x}=2$

Great size

If the function $f(x)$ 满足 $\underset{\begin{array}{c}x\to x_0\\ (x\to \infty )\end{array}}{\lim }f(x)=\infty$ , then$f(x)$ 为 $x\to x_0$(orx $x\to \infty$ ) infinitywhen

注意：$\underset{\begin{array}{c}x\to x_0\\ (x\to \infty )\end{array}}{\lim }f\left(x\right)=\infty$（①式）是 $\underset{\begin{array}{c}x\to x_0\\ (x\to \infty )\end{array}}{\lim }f\left(x\right)=+\infty$（②式）和 $\underset{\begin{array}{c}x\to x_0\\ (x\to \infty )\end{array}}{\lim }f(x)=-\infty$ (formula ③) is the general name. If one of the formulas ②③ is satisfied, it must satisfy the formula ①

It can be seen from the definition that the essence of infinity is a function. The function value of this function approaches infinitely to ∞ during a certain change of the independent variable $\infty$ . Infinity is not a very big number

The relationship between infinitesimal and infinite

In the same change process of the independent variable,

• If $f\left(x\right)$ is infinitesimal, then$\frac{1}{f\left(x\right)}$is infinity
• If $f\left(x\right)$ is infinite, then$\frac{1}{f\left(x\right)}$is infinitesimal

reference

[1] Higher Mathematics, Department of Mathematics, Tongji University, Higher Education Press, Volume 1