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Math Analysis Limits of Functions
Itake O - E Ix-El-O-EI l elsiux-Siu2. Math Analysis Limits of Functions 7 of 41Limitofconvezgentfunctionsfortt112E boundedn o t upperc pi t o }hE ic HeRlthen, Lim fan is s oE=X t o Ital-elseF I IE t h a t x >sucheLimitofconvezgentfunctionsfortt112E lower boundedn o tc pi l - o }E ic HERlthen, Lim fan isE= s ox - a Ifk) - f l e eF I IE t h a t x esuchc-I t possiblenot account thet a k e intot oi s Ix-Ile es since E - t ao r xexpression a INDETERMINATE)(oo-• FORM2. Math Analysis Limits of Functions 8 of 41Limit divergent f u n c t i o n sofPositivelydivergentR,E RE f :IE, EC E →him fix) ift o o=I× →FM FotoR t h a tsuchE far)>Ix-Elt ME, oo r× eForestapi ( n o t{ to }E bounded 1u p p e rc i Limthen, t h e Ri ffelt t o oX t o o→F i re IE that Te x t >X > Ms u c hF o r tpi fo} (not bounded)E lowerc i Limthen, t h e Ri tfelt t oN-FREE Ithat Te x t >X i Ms u c h2. Math Analysis Limits of Functions 9 of 41NegetivelydivergentIR, RE IE, f:E E Ec
Analysis Limits of Functions 10 of 41 Topology Neighbourhoods Let the NEIGHBOURHOOD SPHERICAL E of radius the set is (XER o) III. E-d) s i x e x t -- (E-O, E t o )= xxt . i e I t oI - 0 Definition R IRV - Let x. A neighbourhood of x is any subset of that contains a spherical neighbourhood of x, with radius greater or lesser than . - Each spherical neighbourhood is a subset of the neighbourhoods with greater radius and contains those with a lesser one. f',Xto'x-d' J(E) UI I I. ol c d > XtoX-8 for os o m eo By changing a family of neighbourhoods is obtained. {VUe- I E}, IR V i s neighbourhood o facUx said OfFA M I LY NEIGHBOURHOODS Iis O FAnalysis Limits of Functions 11 of 41
Reeki) An arbitrary union of neighbourhoods of x is a neighbourhood of x .- -I, ÷ )t-t. ill-l, Ellee . g. -if -An arbitrary intersection of neighbourhoods of x is not aneighbourhood of x .-D f u rI-t.nl-lo}up,e . g.However a finite intersection of neighbourhoods is still aneighbourhood.
Definitioni) RV c saids e t i se eNEIGHBOURHOOD O F i tift - contains a( I , too)half kindl i n e theof÷rife RV c saids e t i s eNEIGHBOURHOOD O F - - i tif contains aI)to,half kindl i n e theof÷2.
Math Analysis Limits of Functions 12 of 41
Accumulation ( t o p o l o g i c a l )p o i n tlet R, AC C U M U L AT I O NE POINTE saidis e uCOf E ifHVE f{I}U n E tUt fQu!tion: E Uhow many points of fall in a given neighbourhood of x, ?=Answer: infinitely many!Reineck The concept of limit is a local notion, as it depends only on thebehaviour of the function f in a neighbourhood of x , and it is-therefore insensitive to what happens far from x .
Initio2. Math
(topological definition)
Let I, E, and F be sets. A function f : I → F is said to have a limit L at a point x if for every neighborhood V of L, there exists a neighborhood U of x such that for all t in U - {x}, f(t) is in V. Equivalently, f has a limit L at a point x if the image of every neighborhood of x under f is eventually contained in every neighborhood of L.
(REDUCTION OF LIMITS THROUGH SEQUENCES)
Let I, R, E be sets, and let f : I → R be a function. Then, f has a limit L if and only if for every sequence {Xn} in I that converges to x, the sequence {f(Xn)} converges to L.
(by contradiction)
Assume, by definition, that the limit of f at x is L. Let V be a neighborhood of L. Then, there exists a neighborhood U of x such that for all t in U - {x}, f(t) is in V. Consider a sequence {Xn} in I that converges to x.
text =l - l 'then,Proof {Xu} {II, IEconsider XuE ' →ltex tfiff =L,sincethen lf k n )by LT S , A=L',tex tfixsince l'then f k n )by LT S , i tby l i m i t sequences,ofo fu n i q u e n e s st e l l2. Math Analysis Limits of Functions 18 of 41-µt+ t oX1 × 1:s i g ne . g. If}Limo s i g n a len-ut (en)o s i g n ti I 1→=but-Into (bn) - 1s i g n →= - I(en)l i m ( b u tF l i ms i n c e sign (x)Liontheir does existn o tsign by LT S2. Math Analysis Limits of Functions 19 of 41Theorem ( p e ¥ # u 1tiff F V Elfly UE t h a tsuchoi=the ht},U n E t SCH> oProof Effsince fix) - l theF V E {El,U nUE E lt h a ts u c hl - E h tTCH i Ett a k e E - l L t lfaitl - l e 2 Lf (x) To rthen, furl t oCorkeryf(x)> in neighbourhood Eo faothen, fizz tiff fatsoit far) e x i s t s2. Math Analysis Limits of Functions 20 of 41Theorem ( d o u b l e c o m p e z i )IR,E JEIc E ,Rf, Eh : → t u e tsuchg,Vx (x)f (x)(x) hE,locally I IE g Rlif l i m 41×1 Cx)limy
E==textfine = LProof {Xu} belet arbitrary sequencea n{Xu} LE},E lt h a t Is u c h E X u →(Xu) (Xu)flku) 4by I Iassumption gfl e l(Xu)by h i s tby d o u b l e comparison o f sequencesw ethat LIE f (x) - l2t h e n , by i t s2. Math Analysis Limits of Functions 21 of 41(comparison)Theoremi) SEQUENCES Xu X u # t oYulet It oya → ,Prot t oYu → ,75 KFk> Yu>such t h a t ita >o , KXu XuI sYu → t o oii) FUNCTIONSIR, R,Elet AE,f : Ec E Eg, →lim fa r ) t o=Ix → Vx fizz 8 1 4 - tfly get EI E •Proof {E)E lXu IE Xu →,by f (xu)L T S , t o oi t galyardf e n l e T H E I t oby double girl#tooc o m p a r i s o n ,2. Math Analysis Limits of Functions 22 of 41Remerkeblefunctiousies thed i r e c t continuityofc o n s e n t i e n c eelementary f u n c t i o n s ,of t h e i rbehaviour e a s i l y X # I .defined forisThen, f)(a IRfiff x" I → , doinE EE=o . ,IR( e e f)e x - e t , doinfix E E,o
..fi#eogex=eoset.fEEo1mIY-o.eimsiux--siux ( F e d o m f ),I→x t r u e f o rs u c h properties a r eall elementaryt h e functions2. Math Analysis Limits of Functions 23 of 41Remerkebeelimitsefiygsi.IQ1 f#±i:÷÷::::::.÷:b-h-IDef's Iz1. s i n x .== IDOE, t e n X .
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