Math Analysis I - Limits of Functions Part 2 (Sequences)

Math Analysis Limits of Functions

F E E INh e m e l y , t h a tsuch h i n t an> oiifen ↳ - o o{on} e v e n t u a l l ythen, negativeisF E E INh e m e l y , t h a tsuch h i t # e u r oProof: bydefinitionofdivez.ge#ey-ueucei) IR,K E K isuppose oF rthen t h a t i n e n i ks u c h nif IR, KKE t osupposeFiithen Kt h a t i ns u c h eyeu2.

Theorem ( d o u b l e c o m p a r i s o u r s e u d w i "Faenarimen"Given {bn}{on},the sequences bne n dis o n et ot ie nthen, b n # t o (biffs)the t r u e o n e - 0soreiss a m eProofl e t k i o arbitraryDsince l , ttFa Kt h a t it,s u c h o n >n >b nsuppose I b n ,Fitz buzen252such T h e t n(Fritz)take it m e x= bnthen, Ki n Z e n >n divergenceby d e f i n i t i o n o f b u t t o2.

Theorem AO)( l i m i tThe limit of the sum of concordant divergent sequences, divergestowards infinity (positive or negative):b u ttoo,Qu ± o rthen, bncu Ie a t →= 0

Proof too)(consider i to nE Rlet K, a r b i t r a r ym1 since t o ote nFF, it,t h a ts u c h me n >u >bn2- since a t aFitz by>52t h a t K-ms u c h u >take (5,, 52)it m e x= h t tthen, k-mlb u ti n Cnu o u t=bn by definitioncu o ot= i ee a t divergenceofRemerke: buif a n d I n' →ooIa n ↳ must studiedbeCu bne a t= byc a s e c a s e2. Math Analysis Limits of Functions 37 of 65Theorem BO)( l i m i tThe limit of the sum of two sequences respectively divergent andconvergent, diverges towards infinity (positive or negative):IRkb u ttoo,Qu ' → Ethen, bncu Ie a t →= 0 0Proof too)( c o n s i d e r e n ↳ RKbn Esuppose ' →then, s i n c e a n # t oFit Kit e n >t h e ts u c h snthen c u bn t o o' →= te ,Coroleery{en}let boundedbe and{bn} be positively negativelyo rdivergentbn positivelycu en negativelyist= o rdivergent2. Math Analysis Limits of Functions 38 of 65Theorem @)( l i m i t e dThe limit of the product of divergent sequences,

The limit of the ratio of a divergent sequence is zero.

H E Rbut>o n # t o oc u - B Ithen, ouQuProoslet arbitraryE s o a n d a n # t osupposet o os i n c e →1 e n ,- t o75, k' b e specifieda a n yt h a tsuch n > latersince K,2 - b u t t oE' b e specifiedIbn-KITFitz T h e tsuch laterE'butE'by K- K 5 tcomparisonE'tb y , kcu = \ -K'e n ( 2E - e l U'E' andK 2 kchoose == tKX2{µt#=cu-bye 7yd e=e n on-II.ICal by d e f i n i t i o n limitEr o f o→2.

Math Analysis Limits of Functions 42 of 65P-uetheth.IOsince en#O, t o be|leleuFit Especisiedi tt h a ts u c h u > leterKI t # E 'u h (I)"E" eekE'chose =the nitethen, e lentiei t - I t ] , turnfixing2.

Math Analysis Limits of Functions 43 of 65HierezchyofinfinitiesA sequence is said divergent if it tends to infinity (either positive ornegative). However, different divergent sequences may approach infinitywith different

speeds and an be classified by virtue of such property.

The behaviour of sequences diverging towards ∞ is analogous to those approaching ∞, which can be studied by using the algebra of limits.

The study of the hierarchy of infinities can turn out to be useful when dealing with indeterminate forms of the type:

{X_n} → ∞

{Y_n} → ∞

where X_n ≠ Y_n for all n ≥ N.

Consider a list of significant sequences diverging to infinity:

let {e_n} be a list of divergent positive sequences

then, list E⁺ = {X_n | X_n → ∞}

study different cases that may occur:

1. if the limit of the sequence is 1, then the sequence is bounded
2. if the limit of the sequence is > 1, then the sequence is unbounded
3. if the limit of the sequence is < 1, then the sequence converges to 0

Leudeub yIn symbolsoften ¥10¥ # ↳then ⇐ o foIn#oby sandwich,2. Math Analysis Limits of Functions 47 of 65adf.ie. # III. → t oe uProoflet arbitrarye s ia "if 1 = 0 ,I IIby II0sandwich oeffI tu E N± o timesu-Eu2- e - e . r s # == ( h - 1 ) (h-2) I4 •. . . .¥2..... A¥.¥. I 0 . o= =eL -d I e - Aoee terms¥uby s a n d w i c h , o→,2. Math Analysis Limits of Functions 48 of 65' IntoLife, lim I e o ou!u # t oProof u¥±'T1¥ by sandwichit o ft o o ,II t u1 i l± o largeI 5fon a sufficientlyz w i t hn (u-l)(u-2)u! 1n . . .-