Lecture 8: Infinitesimals and Infinite Quantities

8.1 Infinitesimal

8.1.1 Basic Concepts

#####Definition: infinitelysmall

Description: if the function $f(x)$ has zero limit at $x\to x_0($ or $x\to\infty)$, then $f(x)$ is called an infinitesimal at $x\to x_0($ or $x\to\infty)$.

8.1.2 Comparison of infinitesimals

-Iso-orderinfinitesimal: $\alpha(x)\text{ and }\beta(x)$ The result of division isconstant C(C is not equal to 0);

-Equivalentinfinitesimal: $\alpha(x)\text{ and }\beta(x)$ divide byconstant 1;

• Higher-order infinitesimal: $\alpha(x)\text{ and }\beta(x)$ divide by 0; can be denoted as: $\alpha(x)=o(\beta(x))$
• Low-order infinitesimals: $\alpha(x)\text{ and }\beta(x)$ divide by the opposite order is infinite;
• $\text{ If }\lim\frac{\alpha (x)^{\color{red}{}}}{\left[\beta (x)\right]^{k}\color{red}}=C\neq 0,\text{ Said }$ α is the k-th order infinitesimal of β;

8.1.3 Properties of infinitesimals

-Property 1:The sum of infinitesimalsis still infinitesimal;

-Property 2:The product of finiteinfinitesimals is still infinitesimal;

-Property 3:The product of infinitesimals andbounded quantitiesis infinitesimal;

8.2 Infinity

8.2.1 Basic Concepts

#####Definition: infinitelylarge

Description: if the function $f(x)$ has an infinite limit at $x\to x_0($ or $x\to\infty)$, then $f(x)$ is called an infinite quantity at $x\to x_0($ or $x\to\infty)$.

Explanation

• For every $M>0$, there is a $\delta>0$ such that $0<|x-x_0|<\delta$ implies $|f(x)|>M$.

8.2.2 Comparison of Common Infinity

Concept: Function limit

• When x approaches infinity:
• $$\ln^{\alpha}x<<x^{\beta}<<a^{x}$$
• Among them, $\alpha>0,\beta>0,a>1$
• Example:
• So $\lim_{x\to+\infty}\frac{\ln x}{x}=0$;

Concept: Sequence limits

• Sequence limit: $n\to\infty$
• $$\ln^\alpha n<<n^\beta<<a^n<<n!<<n^n$$
• Where $\alpha>0,\beta>0,a>1$;

8.2.3 Properties of Infinity

-Property 1:The sum of finite positive infinity is infinity;

-Property 2:Theproductof finite infinity is still infinite;

-Property 3:The sum of infinite quantities andbounded variablesremains infinitely large;

8.2.4 Infinite Quantity and Unbounded Variables

• 1. A sequence $\{x_n\}$ tends to infinity in magnitude if, for every $M>0$, there is an $N$ such that $n>N$ implies $|x_n|>M$.
• $|x_n|$ All values after N are very large;
• 2. $\text{ Sequence }\left\{x_n\right\}\text{ It is an unbounded variable }$：$\forall\boldsymbol{M}>\boldsymbol{0},\exists N>\boldsymbol{0},\text{ Envoy }|x_N|>\boldsymbol{M}$
• $|x_n|$ There exists a value at n greater than M, but not necessarily many terms are greater than M;
• Example: $x_n=\begin{cases}n,&n\text{ which is an odd number }\\\mathbf{0},&n\text{ and the number is even }\\\end{cases}$ is an unbounded variable, but not infinite;
• 1 can be inferred from 2, 2 cannot be deduced from 1;
• $\text{ Infinite and vast }\Rightarrow\textbf{ \text{ Unbounded variable }}$

8.3 The Relationship Between Infinity and Infinitesimal

• In the same limiting process, if $f(x)$ is infinite, then $1 / f(x)$ is infinitesimal;
• In the same limiting process, if $f(x)$ is infinitesimal and $f(x)$ is not zero, then $1 / f(x)$ is infinite;