SOME REMARKS ON HORIZONTAL, SLANT, PARABOLIC AND POLYNOMIAL ASYMPTOTE
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1 D I D A C T I C S O F A T H E A T I C S No (4) 3 SOE REARKS ON HORIZONTAL, SLANT, PARABOLIC AND POLYNOIAL ASYPTOTE Tdeusz Jszk Abstct I the techg o clculus, we cosde hozotl d slt symptote I ths ppe the de o symptote o ucto s epded to polyomls Thee e gve omuls o coecets o the multoml symptote o the ucto d some emples o pbolc d cubc symptote Keywods: Hozotl symptote, slt symptote, pbolc symptote, cubc symptote, multoml symptote, tol ucto DOI: 56/dm34 Hozotl d slt symptote The tem symptote mes usully stght le, thus le l s symptote to cuve the dstce om pot P to the le l teds to zeo s P teds to ty log some ubouded pt o the cuve I the cuve s the gph o el ucto ths deto cludes vetcl, hozotl y, d slt symptote y (Clphm 996; Kudvcev 973) It s kow tht the hozotl symptote o ucto y hs ts pmete lm, t s vestgted s, the slt symptote hs ts pmetes: lm d lm Tdeusz Jszk Deptmet o themtcs d Cybeetcs, Wocłw Uvesty o Ecoomcs, Komdosk Steet 8/, Wocłw, Pold E-ml: tdueszjszk@uewocpl
2 46 Tdeusz Jszk I the cse the esult c be deet Fucto y ct hs two deet symptotes y d y The sme s the cse o slt, pbolc d polyoml symptote The ede cetly kows my emples o the ucto wth hozotl o slt symptote Emple The most smple emple o ucto whch hs the hozotl symptote y s y Emple The sum: () = cost + / hs the hozotl symptote y = cost Emple 3 The sum o the le ucto y b d the hypebol om emple s tol ucto d ts slt symptote s the le ucto: y : Souce: ow elboto Fg A tol ucto wth the slt symptote
3 Some emks bout hozotl, slt, pbolc d polyoml symptote 47 Emple 4 Fucto s s ot tol d hs the hozo- tl symptote y : Fg The ucto osclltes close d close oud ts hozotl symptote Souce: ow elboto s Emple 5 Let b whee Ths ucto osclltes close d close oud ts slt symptote y b Pbolc symptote The tem pbolc symptote s deed smlly (Jszk ) Deto A pbol y s pbolc symptote o the ucto y, s, I the cse lm the esult c be deet ()
4 48 Tdeusz Jszk Theoem Let ucto be gve y tevl m,, d hve the pbolc symptote deed by (), the thee est thee lmts: lm, () lm b, (3) c lm, (4) d the pmetes,, e equl:, b, c Poo The equltes () d (4) e equvlet s c Let ε be postve umbe By omul () thee ests umbe such tht o ech > equlty (5) whee, holds The equlty (5) c be dvded by >, the the sequece o equltes holds, hece, the equlty bove poves tht e lm thus the lmt (3) ests, d lm b (6), (7)
5 Some emks bout hozotl, slt, pbolc d polyoml symptote 49 Now we eed to pove tht the equlty () holds, d c Let, by (7) thee ests umbe such tht o ech > double equlty holds, whee Becuse the equlty below s tue: d, Now the bove equlty s dvded by > : We hve (8) The double equlty (8) mples the equlty () whee The poo s complete Coolly The ucto y hs the pbolc symptote, s, d oly the omuls (), (3), (4) hold; the pbol s gve by omul y b c Coolly Fucto hs t most oe pbolc symptote The poo c lso be mde dectly by deto I the omul () holds d the equlty c c c lm holds too, the the ddto o the let sde o () d (9) hs the lmt equl to zeo, e (9)
6 5 Tdeusz Jszk so c c c lm c c lm c, () ; () the omul () poves tht c, om hee t ollows tht the omul () s equvlet to e c c lm lm c c () ; (3) the omul (3) etls tht c, d the omul () s equvlet to lmc e c The dect poo o coolly s complete Emple 6 Let tol ucto be gve by the ollowg omul: Its decomposto hs the om: The pbolc symptote o s deed by y 6 wth the vete the pot 3, y Fucto hs the om: 3 The tble o sgs o helps to mde the plot o the ucto ; t us bove the -s o the tevls d below t o the tevls, ;, ; 3,, d, 3
7 Some emks bout hozotl, slt, pbolc d polyoml symptote 5 Smlly, t s locted bove ts pbolc symptote o the tevl,, d below o, I Fgue 3 thee s peseted gph o ucto, d ts pbolc symptote Souce: ow elboto Fg 3 A tol ucto wth ts pbolc symptote Emple 7 The ucto gve by the omul g s 6 hs the sme pbolc symptote s emple 6; s The gph o t s susod whch us close d close oud the pbol y 6 3 Polyoml symptote Smlly to pbolc symptote, thee s deed the tem polyoml symptote Deto A polyoml p (4)
8 5 Tdeusz Jszk s sd to be polyoml o multoml symptote o the ucto y, s, the equlty holds I the cse lm p the esult c be deet (5) Theoem Let polyoml p gve by (4) be multoml symptote o ucto d the dom o ucto cludes y tevl m, The thee ests + lmts: d so o d so o d o ech lm lm b, (6) b, (7) k lm k bk, (8) lm lm,, the equlty b holds b, (9) b, () Poo The poo s mde by ducto The st ducto step s tvl becuse the equltes (5) d () e equvlet whe b Secod ducto step: let the omul (8) be tue o ech k,,,, d b o ech,,, whee s tul umbe: such tht It eeds to be poved tht
9 Some emks bout hozotl, slt, pbolc d polyoml symptote 53 lm k () By the ducto ssumpto the omul lm () holds Let, by () thee ests k such tht o ech the double equlty below s tue:, whee The equlty bove c be bodeed: The lst equlty s dvded ow by : tht s ; the equlty bove poves tht
10 54 Tdeusz Jszk k lm k k (3) o ech k,, It s ecessy to pove tht Let t ow be oted lm lm (4) (5) Fo ech thee ests such tht o ech, d o the equlty holds Hece the equlty below holds too: (6) (7) Fo the cocluso o the poo, we eed to dvde (7) by d so the equlty (4) holds : (8) Coolly 3 Fucto hs t the most oe polyoml symptote The coecets o the polyoml e gve by the omuls (6)-() Theoem 3 Suppose the ucto y hs the polyoml symptote y p whch s gve by (4) The c be epeseted by the sum p (9)
11 Some emks bout hozotl, slt, pbolc d polyoml symptote 55 whee lm (3) The polyoml p s sd to be the pcpl pt o ucto, wth espect to the set o multomls, d the emde pt o t Poo By ssumpto the equlty (5) holds, ucto s gve by the omul om hee p the poo s complete whee Theoem 4 Let P d p (3) p p ; (3) be tol ucto: P Q (33) Q e polyomls The hs decomposto: P p (34) Q whee p d P e polyomls too, d the degee o P s less th Q The polyoml p s the pcpl o tege pt o the tol ucto P d s the emde o ctol pt o t Q Emple 8 The ucto s tol d hs the decomposto The polyoml y 3 4 s the multoml symptote o the ucto It s ot dcult to mke gph o the ucto d ts symptote
12 56 Tdeusz Jszk Souce: ow elboto Fg 4 A tol ucto wth cubc symptote Reeeces Clphm C (996) themtcs Ood Jszk T () Fukcje wymee Wydwctwo Akdem Ekoomczej we Wocłwu Kudvcev LD (973) атематический анализ Высшая школа oskw
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