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Question Number 155331 by zakirullah last updated on 28/Sep/21
what is limit? also where we use it.
$$\mathrm{what}\:\mathrm{is}\:\mathrm{limit}?\:\mathrm{also}\:\mathrm{where}\:\mathrm{we}\:\mathrm{use}\:\mathrm{it}. \\ $$
Answered by TheHoneyCat last updated on 29/Sep/21
The idea of limit is very very general. It is present in post fields of math (all of the non−descreet ones, but also some others) The general idea of limits is “Looking at things that get closer to a given value with time” This is the case for a squence whose limit is a_∞ lim a_n =a_∞ is not a formula, but a statement defined as folows: ∀d>0 ∃N ∀n>N (−d)<a_n −a_∞ <d which means that for evey distance you want to consider (∀d>0) at some point all the values of the seqence (∃N is that point ∀n>0 are the values that folow) will be closer to a_∞ than d. this of course can be generalised for any kind of space you can imagine (in topology) if you have a distance function d (such that d(x,y) is the distance between x and y) we get ∀ε>0 ∃N ∀n>N d(a_n ,a_∞ )<ε also, it dosn′t have to be a sequence let f be a function from a space with a distance d_1 to another with a distance d_2 lim_(x→x_0 ) f(x)=y_0 means : ∀ε_2 >0 ∃ε_1 >0 d_1 (x,x_0 )<ε_1 ⇒ d_2 (f(x),y_0 ) still the same idea, if you go close enought to x_0 the function will go close enought to y_0 That is a limit and since it is very general, it is used almost everywere in both math and physics but If you want some examples here are a few (note that usualy the distance d(x,y) is ∣x−y∣) this defines what a continious function is indeed, a function is continuous on X when ∀x_0 ∈X lim_(x→x_0 ) f(x)=f(x_0 ) this is also how you define the derivative f ′(x_0 )=lim_(x→x_0 ) ((f(x)−f(x_0 ))/(x−x_0 )) hence how you define speed, and acceleration wich are the fundamentals of physics You can do infinite sums witch is essential in may calculations (it assures you that things don′t blow up) I hope this answers your question also, I skiped over this because it is technicaly not the same thing, but in the case of functions that have values in R (so not every function) you can define “lim_(x→+∞) ” and also “=+∞” it is a bit different since ∞ is not a number and you are never “close to it” but the idea is just to switch the “getting close yo” with a “is never bounded” I live it yo you as an exercise if you want. :−)
$$\mathrm{The}\:\mathrm{idea}\:\mathrm{of}\:{limit}\:\mathrm{is}\:\mathrm{very}\:\mathrm{very}\:\mathrm{general}. \\ $$$$\mathrm{It}\:\mathrm{is}\:\mathrm{present}\:\mathrm{in}\:\mathrm{post}\:\mathrm{fields}\:\mathrm{of}\:\mathrm{math} \\ $$$$\left(\mathrm{all}\:\mathrm{of}\:\mathrm{the}\:\mathrm{non}−\mathrm{descreet}\:\mathrm{ones},\:\mathrm{but}\:\mathrm{also}\:\mathrm{some}\:\mathrm{others}\right) \\ $$$$ \\ $$$$\mathrm{The}\:\mathrm{general}\:\mathrm{idea}\:\mathrm{of}\:\mathrm{limits}\:\mathrm{is}\:“\mathrm{Looking}\:\mathrm{at}\: \\ $$$$\mathrm{things}\:\mathrm{that}\:\mathrm{get}\:\mathrm{closer}\:\mathrm{to}\:\mathrm{a}\:\mathrm{given}\:\mathrm{value}\:\mathrm{with}\:\mathrm{time}'' \\ $$$$ \\ $$$$\mathrm{This}\:\mathrm{is}\:\mathrm{the}\:\mathrm{case}\:\mathrm{for}\:\mathrm{a}\:\mathrm{squence}\:\mathrm{whose}\:\mathrm{limit}\:\mathrm{is}\:{a}_{\infty} \\ $$$$\mathrm{lim}\:{a}_{{n}} ={a}_{\infty} \:\mathrm{is}\:\mathrm{not}\:\mathrm{a}\:\mathrm{formula},\:\mathrm{but}\:\mathrm{a}\:\mathrm{statement} \\ $$$$\mathrm{defined}\:\mathrm{as}\:\mathrm{folows}: \\ $$$$\forall{d}>\mathrm{0}\:\exists{N}\:\forall{n}>{N}\:\:\:\left(−{d}\right)<{a}_{{n}} −{a}_{\infty} <{d} \\ $$$$\mathrm{which}\:\mathrm{means}\:\mathrm{that}\:\mathrm{for}\:\mathrm{evey}\:\mathrm{distance}\:\mathrm{you}\:\mathrm{want} \\ $$$$\mathrm{to}\:\mathrm{consider}\:\left(\forall{d}>\mathrm{0}\right)\:\mathrm{at}\:\mathrm{some}\:\mathrm{point}\:\mathrm{all}\:\mathrm{the}\: \\ $$$$\mathrm{values}\:\mathrm{of}\:\mathrm{the}\:\mathrm{seqence}\:\left(\exists{N}\:\mathrm{is}\:\mathrm{that}\:\mathrm{point}\:\right. \\ $$$$\left.\forall{n}>\mathrm{0}\:{are}\:{the}\:{values}\:{that}\:{folow}\right)\:\mathrm{will}\:\mathrm{be}\:\mathrm{closer} \\ $$$$\mathrm{to}\:\mathrm{a}_{\infty} \:\mathrm{than}\:\mathrm{d}. \\ $$$$ \\ $$$$\mathrm{this}\:\mathrm{of}\:\mathrm{course}\:\mathrm{can}\:\mathrm{be}\:\mathrm{generalised}\:\mathrm{for}\:\mathrm{any}\: \\ $$$$\mathrm{kind}\:\mathrm{of}\:\mathrm{space}\:\mathrm{you}\:\mathrm{can}\:\mathrm{imagine}\:\left({in}\:{topology}\right) \\ $$$$\mathrm{if}\:\mathrm{you}\:\mathrm{have}\:\mathrm{a}\:\mathrm{distance}\:\mathrm{function}\:{d}\:\left(\mathrm{such}\:\mathrm{that}\:\right. \\ $$$$\left.{d}\left({x},{y}\right)\:\mathrm{is}\:\mathrm{the}\:\mathrm{distance}\:\mathrm{between}\:{x}\:\mathrm{and}\:{y}\right)\: \\ $$$$\mathrm{we}\:\mathrm{get}\: \\ $$$$\forall\varepsilon>\mathrm{0}\:\exists{N}\:\forall{n}>{N}\:{d}\left({a}_{{n}} ,{a}_{\infty} \right)<\varepsilon \\ $$$$ \\ $$$${also},\:{it}\:{dosn}'{t}\:{have}\:{to}\:{be}\:{a}\:{sequence} \\ $$$$\mathrm{let}\:{f}\:\mathrm{be}\:\mathrm{a}\:\mathrm{function}\:\mathrm{from}\:\mathrm{a}\:\mathrm{space}\:\mathrm{with}\:\mathrm{a}\:\mathrm{distance} \\ $$$${d}_{\mathrm{1}} \:\mathrm{to}\:\mathrm{another}\:\mathrm{with}\:\mathrm{a}\:\mathrm{distance}\:{d}_{\mathrm{2}} \\ $$$$\mathrm{lim}_{{x}\rightarrow{x}_{\mathrm{0}} } {f}\left({x}\right)={y}_{\mathrm{0}} \\ $$$${means}\:: \\ $$$$\forall\varepsilon_{\mathrm{2}} >\mathrm{0}\:\exists\varepsilon_{\mathrm{1}} >\mathrm{0}\:\:\:{d}_{\mathrm{1}} \left({x},{x}_{\mathrm{0}} \right)<\varepsilon_{\mathrm{1}} \Rightarrow\:{d}_{\mathrm{2}} \left({f}\left({x}\right),{y}_{\mathrm{0}} \right) \\ $$$$\mathrm{still}\:\mathrm{the}\:\mathrm{same}\:\mathrm{idea},\:\mathrm{if}\:\mathrm{you}\:\mathrm{go}\:\mathrm{close}\:\mathrm{enought}\:\mathrm{to}\:{x}_{\mathrm{0}} \\ $$$$\mathrm{the}\:\mathrm{function}\:\mathrm{will}\:\mathrm{go}\:\mathrm{close}\:\mathrm{enought}\:\mathrm{to}\:{y}_{\mathrm{0}} \\ $$$$ \\ $$$$\mathrm{That}\:\mathrm{is}\:\mathrm{a}\:\mathrm{limit}\:\mathrm{and}\:\mathrm{since}\:\mathrm{it}\:\mathrm{is}\:\mathrm{very}\:\mathrm{general},\:\mathrm{it} \\ $$$$\mathrm{is}\:\mathrm{used}\:\mathrm{almost}\:\mathrm{everywere}\:\mathrm{in}\:\mathrm{both}\:\mathrm{math}\:\mathrm{and} \\ $$$$\mathrm{physics} \\ $$$$ \\ $$$$\mathrm{but}\:\mathrm{If}\:\mathrm{you}\:\mathrm{want}\:\mathrm{some}\:\mathrm{examples}\:\mathrm{here}\:\mathrm{are}\:\mathrm{a}\:\mathrm{few} \\ $$$$\left(\mathrm{note}\:\mathrm{that}\:\mathrm{usualy}\:\mathrm{the}\:\mathrm{distance}\:{d}\left({x},{y}\right)\:{is}\:\mid{x}−{y}\mid\right) \\ $$$$\mathrm{this}\:\mathrm{defines}\:\mathrm{what}\:\mathrm{a}\:\mathrm{continious}\:\mathrm{function}\:\mathrm{is} \\ $$$$\mathrm{indeed},\:\mathrm{a}\:\mathrm{function}\:\mathrm{is}\:\mathrm{continuous}\:\mathrm{on}\:\mathrm{X}\:\mathrm{when} \\ $$$$\forall{x}_{\mathrm{0}} \in{X}\:\mathrm{lim}_{{x}\rightarrow{x}_{\mathrm{0}} } {f}\left({x}\right)={f}\left({x}_{\mathrm{0}} \right) \\ $$$$ \\ $$$$\mathrm{this}\:\mathrm{is}\:\mathrm{also}\:\mathrm{how}\:\mathrm{you}\:\mathrm{define}\:\mathrm{the}\:\mathrm{derivative} \\ $$$${f}\:'\left({x}_{\mathrm{0}} \right)=\mathrm{lim}_{{x}\rightarrow{x}_{\mathrm{0}} } \:\frac{{f}\left({x}\right)−{f}\left({x}_{\mathrm{0}} \right)}{{x}−{x}_{\mathrm{0}} } \\ $$$$\mathrm{hence}\:\mathrm{how}\:\mathrm{you}\:\mathrm{define}\:\mathrm{speed},\:\mathrm{and}\:\mathrm{acceleration} \\ $$$$\mathrm{wich}\:\mathrm{are}\:\mathrm{the}\:\mathrm{fundamentals}\:\mathrm{of}\:\mathrm{physics} \\ $$$$ \\ $$$$\mathrm{You}\:\mathrm{can}\:\mathrm{do}\:\mathrm{infinite}\:\mathrm{sums}\:\mathrm{witch}\:\mathrm{is}\:\mathrm{essential}\:\mathrm{in} \\ $$$$\mathrm{may}\:\mathrm{calculations}\:\left(\mathrm{it}\:\mathrm{assures}\:\mathrm{you}\:\mathrm{that}\:\mathrm{things}\:\right. \\ $$$$\left.\mathrm{don}'\mathrm{t}\:\mathrm{blow}\:\mathrm{up}\right) \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$\mathrm{I}\:\mathrm{hope}\:\mathrm{this}\:\mathrm{answers}\:\mathrm{your}\:\mathrm{question} \\ $$$$\mathrm{also},\:\mathrm{I}\:\mathrm{skiped}\:\mathrm{over}\:\mathrm{this}\:\mathrm{because}\:\mathrm{it}\:\mathrm{is}\:\mathrm{technicaly} \\ $$$$\mathrm{not}\:\mathrm{the}\:\mathrm{same}\:\mathrm{thing},\:\mathrm{but}\:\mathrm{in}\:\mathrm{the}\:\mathrm{case}\:\mathrm{of}\:\mathrm{functions} \\ $$$$\mathrm{that}\:\mathrm{have}\:\mathrm{values}\:\mathrm{in}\:\mathbb{R}\:\left(\mathrm{so}\:\mathrm{not}\:\mathrm{every}\:\mathrm{function}\right) \\ $$$$\mathrm{you}\:\mathrm{can}\:\mathrm{define}\:“\mathrm{lim}_{{x}\rightarrow+\infty} ''\: \\ $$$$\mathrm{and}\:\mathrm{also}\:“=+\infty''\: \\ $$$$\mathrm{it}\:\mathrm{is}\:\mathrm{a}\:\mathrm{bit}\:\mathrm{different}\:\mathrm{since}\:\infty\:\mathrm{is}\:\mathrm{not}\:\mathrm{a}\:\mathrm{number}\: \\ $$$$\mathrm{and}\:\mathrm{you}\:\mathrm{are}\:\mathrm{never}\:“{close}\:{to}\:{it}'' \\ $$$$\mathrm{but}\:\mathrm{the}\:\mathrm{idea}\:\mathrm{is}\:\mathrm{just}\:\mathrm{to}\:\mathrm{switch}\:\mathrm{the}\:“{getting}\:{close}\:{yo}'' \\ $$$$\mathrm{with}\:\mathrm{a}\:“{is}\:{never}\:{bounded}'' \\ $$$$\mathrm{I}\:\mathrm{live}\:\mathrm{it}\:\mathrm{yo}\:\mathrm{you}\:\mathrm{as}\:\mathrm{an}\:\mathrm{exercise}\:\mathrm{if}\:\mathrm{you}\:\mathrm{want}. \\ $$$$\left.:−\right) \\ $$
Commented by zakirullah last updated on 29/Sep/21
so great sir! A boundle of thanks.
$$\mathrm{so}\:\mathrm{great}\:\mathrm{sir}!\:\mathrm{A}\:\mathrm{boundle}\:\mathrm{of}\:\mathrm{thanks}. \\ $$
Commented by TheHoneyCat last updated on 29/Sep/21
Your welcome ��

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