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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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