math_ (function & sequence) meaning of limit & misunderstanding and symbol sorting / neighborhood & de centring neighborhood & neighborhood radius

$*$ Meaning of limit & Misunderstanding and symbol sorting

$\ast$ Summary of the definition of sequence and limit of function

Definition of limit & understand $⊳$

sequence limit

$$\begin{gathered} \forall \epsilon >0,\exists N>0, \\ when:(n>N) \\ then:|x_n-a|<\epsilon \end{gathered}$$

$$\begin{gathered} tosetrenIt\; means\epsilon \\ indeedset{\delta }_{\epsilon },sendhave\; toWhenn>N_{\epsilon }(Itsin,nbyjustwholeCount(surfaceinCountColumnOftermOforderNumber)when,namely \\ n\in R_{N_{\epsilon }}=(N_{\epsilon },+\infty )when,ProtectProvecanenoughfullfoot(\epsilon carryOutOfwantseek):|x_n-a|<\epsilon , \\ namelyfullfoot{x}_n\in (a-\epsilon ,a+\epsilon ) \end{gathered}$$

• Then call it a Is a sequence of numbers $x_{n}stayx\to \infty$ The limit of time

• $$\underset{n\to \infty }{\lim }{x}_n=a$$

Neighborhood & De centered neighborhood & Neighborhood radius

$$\begin{gathered} set\; up{x}_0\in \mathbb{R},\delta >0,openDistrictbetween{R}_{\delta }=(x_0-\delta ,x_0+\delta )callby\underline{x_{0}Of\delta adjacentDomain}, \\ spot{x}_{0}Of\underline{Go\; toheart\delta adjacentDomain}rememberdo\mathring{U}(x_0,\delta ), \\ \delta callbyadjacentDomainAnd\; a\; halfpath({\delta }_{radius}) \end{gathered}$$

Function limit

The limit of the function when the independent variable tends to infinity

• The limit of a function is similar to the limit of a sequence of numbers , Especially when $x\to +\infty$ When

The limit of a function when its independent variable approaches a finite value

$$\begin{gathered} \forall \epsilon >0,\exists \delta >0, \\ when:(0<|x-x_0|<\delta ,(namely{x}_{0}OfGo\; toheart\delta adjacentDomain:\mathring{U}(x_0,\delta ))) \\ then:|f(x)-a|<\epsilon ;(namely,aOf\epsilon adjacentDomainU(a,\epsilon ),(NoGo\; toheart)) \\ becall,f(x)\to a(x\to x_0) \end{gathered}$$

Brief version

• The limit of a function whose arguments tend to be finite

$$\begin{gathered} stayx\to x_{0}Oftoochengin \\ tosetrenIt\; means\epsilon \\ savestay(canWithindeedset){\delta }_{\epsilon },sendhave\; toWhenxfullfoot0<|x-x_0|<{\delta }_{\epsilon }when,namely: \\ \underline{x\in R_{{\delta }_{\epsilon }}=(x_0-{\delta }_{\epsilon },x_0+{\delta }_{\epsilon })}when,ProtectProvecanenoughfullfoot(\epsilon carryOutOfwantseek):|f(x)-a|<\epsilon , \\ namelyfullfoot\underline{f(x)\in (a-\epsilon ,a+\epsilon )} \end{gathered}$$

• Then address a by f(x) In the independent variable $xTrendnearOn{x}_0$ The limit value of :

• $$\underset{x\to x_0}{\lim }f(x)=a$$

In the definition of functional limits 4 It's a constant ( Symbol )

• $x_0$: The independent variables x The value to approach

  • The independent variables x May also approach $\infty$

• $\delta$: Help define a specific interval ($R_{\delta }$ or $R_{{\delta }_{\epsilon }}$) The constant

• a: Limit value

• $\epsilon$: Near the limit ( wave ) Allowable value of deviation

  • $\epsilon$ Used to characterize functions f(x) Approach the limit value within a given interval a The degree of ( Any small ( Arbitrarily harsh ) The allowed function value of is a Nearby fluctuation range )

$$\begin{gathered} \delta (by了bodypresentOf\delta and\epsilon AndbetweenOfUnitedsystem,IPeoplecanWithtake\delta surfaceinby{\delta }_{\epsilon }）useOnmomentdraw\underline{,setOfstripPieces\; ofDistrictbetween} \\ (by了SketchStatementFangthen,Districtbetweenrememberby{R}_{\delta }) \\ (AnalogySuch\; asmomentdrawOneindividualadjacentDomain)surfacein,renIt\; meansSmallOf\epsilon allcanenoughphaseShould\; beOfindeedset(look\; forTo)OneindividualfullfootstripPieces\; of: \\ (send,setDistrictbetween{R}_{{\delta }_{\epsilon }}InsideOfLetterCountvaluephaseYesOnextremelylimitaOfwavedynamic(partialmovey=a)OfdistanceleaveNosupertoo\epsilon )Of\epsilon value \end{gathered}$$

The symbolic description of limit and its equivalent mathematical language description

With the limit of the sequence wei’li

• $$\begin{gathered} \underset{n\to \infty }{\lim }{x}_n=a*\forall \epsilon >0,\exists N(N\in N^{+}(justwholeCountSet)), \\ Whenn>Nwhen,fullfoot|x_n-a|<\epsilon \end{gathered}$$

• $$\begin{gathered} \underset{x\to \infty }{\lim }{x}_n=asurfaceinWhennchargebranchBig,x_{n}AndaJustcanWithPick\; upnearTorenIt\; means‘preFirsttoset‘Ofchengdegree \\ namely,|x_n-a|canWithSmallOnrenIt\; meanspreFirsttosetOf\epsilon \\ TheextremelylimitOfanotherOneKind\; ofWriteLawyes{x}_n\to a(n\to \infty ) \end{gathered}$$

  • Be careful , When ε You can take whatever you like ( Small enough ) When , Can reflect the meaning of the limit ( It describes the closeness of the sequence of numbers to the limit in the process close to the limit )

  • N And predetermined $\epsilon$ of , however N No ε Function of ( Because the same ε Can correspond to multiple ( Even an infinite number ) eligible N)

  • A specific function example can be given to assist in understanding and discrimination

  • $$\begin{gathered} x_n=\frac{1}{n};(x_n=0(x\to \infty )singletransferandAndYesextremelylimit0) \\ {x}_n=\frac{(-1{)}^n}{n};(x_n=0(n\to \infty )NosingletransferbutyesYesextremelylimit0) \end{gathered}$$

Understand the mistakes that limit is easy to enter & give an example

The process approaching the limit is different from the process in which the terms are approaching the limit

• Close to the limit $\nLeftrightarrow$ Getting closer to the limit

• $$\begin{gathered} \underset{x_n\to \infty }{\lim }{x}_n=0 \\ IPeopleNocanenoughsay,x_{n}along\; withthen\to \infty ,x_{n}The\; moreCome\; onThe\; morePick\; upnear{x}_n \end{gathered}$$

• $$x_n=\frac{2+(-1{)}^n}{n};(x_n=0(n\to \infty )CountColumnvarioustermallstay>0,AndNosingletransfer,butyesYesextremelylimit0)$$

  • In this case ,$x_1,x_2,x_3,x_4,\dots branchotheretc.On1,\frac{3}{2},\frac{1}{3},\frac{3}{4}$…

  • $$\begin{gathered} x_n=\{\begin{array}{l}\frac{2+(-1)}{n}=\frac{1}{n};n\%2\ne 0(nbyp.Count)\\ \frac{2+1}{n}=\frac{3}{n};n\%2=0(nbyaccidentallyCount)\end{array} \\ cansee,twoindividualSonCountColumnallfullfoot{x}_n\to 0(n\to \infty );andCountColumn{x}_{n}yesVibrationSwingThe\; earthTrendnearOn0 \end{gathered}$$

The geometric meaning of limit