10. Limits involving infinity

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1 . Limits involving infinity It is known from th it ruls for fundamntal arithmtic oprations (+,-,, ) that if two functions hav finit its at a (finit or infinit) point, that is, thy ar convrgnt, th it of thir sum, product, tc. will b qual to th sum, product, tc. of thir its (cpt in cas of a quotint with zro dnominator). In othr words, th opration of finding th it is intrchangabl with th fundamntal arithmtic oprations. In th tabl blow w considr thos cass, whn on of th two functions (or both) tnds to infinity (is propr divrgnt), or th dnominator function tnds to. In such situations two cass ar possibl: ithr th it of th prssion of fundamntal arithmtic can b givn in advanc (dfinit it prssion), or it cannot b givn in advanc in lack of additional information, and th it can b anything or vn may not ist (indfinit it prssion), in th lattr typ of cass th unknown its ar symbolizd by qustion marks. Thorm (it ruls involving infinity) ASSUMPTION STATEMENT SYMBOLIC NOTATION ( If and, thn (????

2 ( c ( c If c and, thn (c ral constant), if c, if c, c, if c c, if c?, if c? c If or c, and, thn (+: and > around, that is, it tnds to from th positiv sid), c, if c If and, thn?? All statmnts of th thorm rmain valid, if if rplacd by th on sidd its, +, or (so in th assumption as in th corrsponding statmnt). Th quations in th symbolic notation must not b considrd as ordinary quations (.g, you must not subtract from both sids, tc.). Th lft sid of th quation symbolizs th typ of th it prssion, th right sid givs th it in dfinit cass or a qustion mark symbolizs uncrtainty in th indfinit cass. Th indfinit prssions of typ / and / can b handld by mans of l Hospital s rul. Th typ. can b rducd to typ / via th transformation f(. = f(/(/ Th diffrnc of typ can b rducd to a quotint typ by bringing to a common dnominator or using othr transformations. Limits of composit functions Thorm. If u and f ( u) L, th uu f ( u f ( f ( u) L. uu u If, in addition, u is finit and f( is continuous at u, thn f ( f ( f ( u)

3 Rmark. Th thorm also holds tru for on-sidd its including its at infinity (,. u,.l can b or ). Without notation w can also writ: u=u f(u)=f(l. Eampl: cot( ) cot( u), u bcaus: u= + cot (u)=cot ( ). L Hospital s rul Thorm. If (typ /) or (typ /), and both functions ar diffrntiabl around, thn providd th lattr it ists. f (, Rmark. Th thorm also holds tru for on-sidd its including its at infinitis ( around mans thn a on-sidd nvironmnt). In addition to th typ / it can also b applid to th typs /, /() and /(), as wll. Important: Always chck th conditions of applicability bfor using th rul s formula! If not usd for th propr typ of prssion, fals rsults ar to b pctd. Eampls: sin cos a) typ /: cos. b) typ. (), chang into typ /, us l Hospital s, thn simplify and substitut: (. c) typ, rduc to quotint, first us composit s and thn l Hospital s rul: (ln( ) (ln( ) ln ) ln ln ln ln Powr prssions: us th ponntial transscription (f( = ln(f( ln = f( and find th it of th formula in th ponnt. For ampl, utilizing th rsult of

4 th prvious ampl b) (substituting u= ln w can find this it: u u. Comparing th rats of convrgnc or propr divrgnc Dfinition. Assum that. Thn w us th following trminology: : f( tnds to fastr (in ordr of magnitud) than, as if c : f( and tnd to in th sam ordr of magnitud (c=: qually fast), if : f( tnds to slowr (in ordr of magnitud) than, as

5 Dfinition. Assum that trminology:. Thn w us th following : f( tnds to fastr (in ordr of magnitud) than, as if c : f( and tnd to in th sam ordr of magnitud (c=: qually fast), if : f( tnds to slowr (in ordr of magnitud) than, as It is actly l Hospital s rul to us for such rat comparisons, bcaus th assumptions ar th sam. For ampl, by th abov ampl a), f ( = sin and = tnd qually fast to as, bcaus thir quotint tnds to c=. PROBLEM LIMITS OF RATIONAL FRACTIONAL FUNCTIONS AT A FINITE POINT Dtrmin th following its: 4 4 a) 4 b) c) d) PROBLEM LIMITS OF RATIONAL FRACTIONAL FUNCTIONS AT INFINITY Dtrmin th following its: a) 6 4 b) (4 ) c) d) 4 ) f) 9 g) ( h) 8 6 PROBLEM LIMITS OF ELEMENTARY FUNCTIONS Giv th its of th lmntary functions (sktch th graph and obsrv th it): a) b) c) d) ) f) g) h) i) j)

6 k) l) m) lg n) o) p) log q) sin r) cos s) tan t) cot u) lg v) tan w) tan 4 cot PROBLEM 4 DEFINITE LIMIT EXPRESSIONS; COMPOSITE FUNCTIONS Dtrmin th following its: a) ( ) b) ( ) c) ( cot tan d) ) f) log,5 g) h) cos i) j) sin ( ) Dtrmin th its of th following composit functions: a) ln( ) b) sin c) d) ) cot g) cos lcos f) tan cos h) ln i) j) PROBLEM 5 INDEFINITE LIMIT EXPRESSIONS: L HOSPITAL S RULE Dtrmin th following its: a) lg b) d) ) ( )cot sin c) sin f) sin cos

7 g) 4 h) i) cos sin j) tg k) sin( cos sin l) ln( m) log n) ln sin cos o) p) q) (cos ) r) PROBLEM 6 Find out th its of th two functions (sktch thir graphs) and compar th rat of thir convrgnc (or propr divrgnc)! a) and ( cos, as b) and, as c) and, as l d) and ln, as ) log and cot, as f) and, as g) and ( ), as h) cot and, as i) tan and, as j) and, as k) cot π and ln, as l) and -, as PROBLEM 7 FINDING LIMITS FOR THE EXAMINATION OF FUNCTIONS Whr is th function continuous (domain)? Dtrmin th (possibly on-sidd) its of th function at th nd-points of its domain! (Attntion: l Hospital s rul is not always applicabl!) a) ( ) b) c) d) ) f) g) h) i) j) m) / k) n) / l) o) ln p) q) cos( ) r)

8 RESULTS. a) b) c) d) dos not ist (lft:, right: ). a) b) c) / d) ) f) g) 4/5 h). a) b) c) d) ) f) dos not ist (not dfind) g) 4 h) i) j) k) l) m) n) dos not ist (not dfind) o) p) q) dos not ist (oscillating) r) s) dos not ist (lft:, right: ) t) u) v) w) 4. a) b) c) d) ) dos not ist (lft:, right: ) f) g) h) i) j) 5. a) b) c) d) ) f) g) h) i) j) 6. a) b) c) d) ) f) g) h) i) / j) k) / l) m) n) / o) p) q) / r) / 7. a) ; slowr b) ; slowr c) ; fastr d) ; fastr ) ; log / slowr f) ; fastr g) ; slowr h) only on-sidd its ist: from th lft, from th right ; qually fast i) only on-sidd its ist: from th lft, from th right ; qual ordr of magnitud (c=) j) ; fastr k) ; qual ordr of magnitud (c=/) l) ; fastr 8. a) D =(,), Lim: both at and at : = b) D =(,), Lim: at : =, at : = c) D =(,)(,)(,), Lim: both at and at : = ; at lft: =, right: = ; at oppositly d) D =(,), Lim: both at and at : = ) D =(,)(,), Lim: at : =, at : =; at : =, at +: = f) D =[,], Lim: both at + and at : = g) D =(,), Lim: at : =, at : = h) D =(,), Lim: both at and at : = i) D =(,)(,), Lim: at : =, at : = ; at : =, at +: = j) D =(,)(,), Lim: both at and at : = ; at : =, at +: = k) D =(,)(,), Lim: both at and at : = ; at : = l) D =(, ), Lim: at +: = ; at : = m) D =(,), Lim: at +: = ; at : = n) D =(,)(,), Lim: at +: = ; at : = ; at : =, at +: = o) D =(,)(,), Lim: both at and at : = ; at : = ; at +: = p) D =(,)(,), Lim: both at, at and at : = q) D =(,), Lim: both at and at : = r) D =(, ), Lim: at +: = ; at : =.

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