After the completion of this section the student

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1 Chapter II CALCULUS II. Liits ad Cotiuity 55 II. LIMITS AND CONTINUITY Objectives: After the copletio of this sectio the studet - should recall the defiitios of the it of fuctio; - should be able to apply the properties ad it theores for evaluatio of its; - should recall the defiitio of cotiuous fuctio ad its properties; Cotets:. Liits - it of a fuctio - oe-sided its - its at ifiity - ifiite its. Liit Theores 3. Eaples 4. Cotiuity 5. Cotiuity Theores 6. Eaples 8. Review Questios ad Eercises 9. Liits ad Cotiuity of Fuctios with Maple

2 56 Chapter II CALCULUS II. Liits ad Cotiuity II. LIMITS We say that L is a it of the fuctio f ( ) whe approaches c ad write f = L if f ( ) becoes arbitrarily close to L whe gets sufficietly close to c. This is just a loose iforal itroductio to the it which requires ore rigorous atheatical defiitio. We will cosider the atheatical cocept of a it where the topological properties of the set of the real ubers play a sigificat role. I this cotet, the ter approach is very iportat. Approach ca be treated as a process of fidig the poits o the real lie which becoe closer ad closer to soe fied poit c. Let < c be a poit left to the poit c. Deote the distace betwee these poits as δ = c. Let us costruct a sequece of poits: = c δ δ = c k= c δk δ k = c k It does ot atter how sall is the distace δ k, we still ca divide it by two ifiitely ay ties, ad we ca approach the poit c as close as we wat. I atheatical laguage it ca be described i the followig way: for ay sall δ > there eists a atural uber K such that c < δ for all k > K k (that costitutes that c is a itig poit of the sequece k, k c). That is why the approach to soe fied poit c we will be described by the distace of a variable poit to the poit c by the iequality: < c < δ It is equivalet to two iequalities c δ < < c ad c< < c+ δ It eas that ca be ay poit which has a distace fro the poit c ot bigger tha δ. This cotiuous approach (ot just of a sequece) we deote by c If we cosider oly the first iequality, the we call it approach fro the left: c δ < < c c The secod iequality costitutes approach fro the right: c< < c+ δ c + I this approach, we did ot allow the poit to coicide with the poit c. If we write for soe fied uber L ad soe ε > that just f L < ε or L ε < f < L+ ε the it eas that variable poit f ca be close to the poit L with a distace saller tha ε, icludig that f ca coicide with the poit L. This laguage of descriptio of the ability of variable poits to be close to soe fied poits was itroduced i the 9 th cetury by Weierstrass ad it is called the ε δ descriptio ad it is used for the foral defiitio of the it of a fuctio, ad it is the iportat tool for rigorous proofs ad derivatio i the oder atheatics. It should be oted, that ituitively the cocept of the it was used by the atheaticias fro the aciet ties (recall Zeo s parado about the Achilles ad a turtle, or Archiedes calculatio of the area eclosed by a parabola by a itig process of addig areas of the fillig triagles), the cocept of the it of the fuctios was fudaetal i the developig of calculus by Newto ad Leibitz i the 7 th cetury without the foral defiitio of the it, which was itroduced alost two hudred years later by Cauchy.

3 Chapter II CALCULUS II. Liits ad Cotiuity 57. DEFINITIONS: Let the fuctio f ( ) be defied o the ope set D, ad let c be a it poit of D (it eas that f ( ) is ot ecessarily is defied at the poit c ). f. (The Liit of a Fuctio ε δ defiitio) We say that L is a it of the fuctio f whe approaches c f = L if for ay ε > there eists a δ > such that < < the if c δ f L < ε. (Oe-Sided Liits) f We say that L is a it of the fuctio f whe approaches c f = L if for ay ε > there eists a δ > such that if c δ c < < the f L < ε We say that L is a it of the fuctio f whe approaches c + f = L + if for ay ε > there eists a δ > such that if c δ c < < the 3. (Liits at Ifiity) f L < ε f We say that L is a it of the fuctio f whe approaches f = L if for ay ε > there eists a B > such that if > B the f L < ε We say that L is a it of the fuctio f whe approaches f = L if for ay ε > there eists a B > such that if < B the f L < ε 4. (Ifiite Liits Vertical Asyptotes) f We say that a it of the fuctio f ( ) is ifiite whe approaches c : f = if for ay B > there eists a δ > such that < < the f = if c δ f > B ad if for ay B > there eists a δ > such that < < the if c δ f < B Oe-sided ifiite its also ca be defied (see Eercise ).

4 58 Chapter II CALCULUS II. Liits ad Cotiuity. LIMIT THEOREMS: Theore. (Heie) (it of fuctios vs. it of sequeces) f ( k ) L Let the fuctio f ( ) be defied o the ope set D, ad let c be a itig poit of set D. The = if ad oly if f L f = L for ay sequece c, D { c} k D c This theore traslates the cocept of the it of fuctios to the laguage of the it of sequeces where the cocept of approach is clear. This stateet ca be used as a defiitio of the it. The proof of this theore is a good eercise for aipulatio with the defiitios. Theore. (eistece of a it) = if ad oly if f L f = f = L c c + I the followig theores we assue the approach c where c ca be c, c +, c, or. Fuctios f ( ) ad least for ( a,b ) { c}, c ( a,b). g are assued to be defied at Theore.3 Theore.4 Theore.5 (uiqueess of the it) If f = L ad (algebra of its) f = M the L = M Let f = L ad g (i) kf = k L k f + g = L + M (ii) f g = L M (iii) f L (iv) = g M f g = f = ideteriat g (it of the coposite fuctio) = M. The if M if L ad M = if L = M = Let f = L ad g = M L g f = M g f

5 Chapter II CALCULUS II. Liits ad Cotiuity 59 y = f f Theore.6 (iverse fuctio theore) L f ( y) = if ad oly if f L f y = c y L c = f y g f Theore.7 (copariso theore) Let f = L ad g if f g = M o ( a,b ), c ( a,b) L M g f h Theore.8 (sadwich theore, squeeze play) Let f = L ad g = L if f h g o ( a,b ), c ( a,b) h = L Theore.9 (L Hospital s Rule resolvig of the ideteriate or ) Let f ( ) ad ( a,b ) { c }, ad let g for all ( a,b ) { c} Let f = ad g be defied ad differetiable o or f = ad g. g = (ideteriate ) = (ideteriate ) The a) if b) if f g f g ( ) = L the = the f g f g = L = Theore. (cotiuous fuctio theore) f g Let the fuctio g( ) be cotiuous at f eists, the = L, ad c g L ( L) = g f g f g f

6 6 Chapter II CALCULUS II. Liits ad Cotiuity 3. EXAMPLES: ) Calculatio of the its usig defiitio Graphical ethod Graphical visualizatio of the defiitio of the it shows that to satisfy the coditios of the defiitio: for fied value of ε > we have to fid the correspodig δ > Fro the cotet of the it defiitio it is clear that that ubers ε > ad f i the δ > are sall, so let the be such that variatio of the fuctio eighborhood of the poit c is sall. Poits L + ε ad L ε o the y -ais defie the iterval where should be the values of the fuctio f ( ). If the fuctio f o-syetric iterval ( c δ,c δ) syetric iterval ( c δ,c + δ) with δ i { δ, δ } is o-liear the i geeral + satisfies this coditio, but the saller = will also satisfy the defiitio of the it. Therefore, first solve equatios: ( δ ) ( δ ) f c + = L + ε f c = L ε the choose δ i { δ, δ } = to satisfy the defiitio of the it. f = Usig defiitio of the it show that = 4 Idetify: f =, c =, L 4 =. Fi ε >. Set up equatios: ( δ ) + = 4+ ε δ = + + ε > ( δ ) = 4 ε δ = ε > The δ i { δ, δ } = = + + ε So, if ( + + ε ) < < ad < < + ( + + ε ) the for ay ε > 3 + ε < < ad < < + + ε < < + + ε 4 ε 4 < < + ε ad, therefore, accordig to defiitio = 4. More geeral proble, c c = c, yields δ = + + ε. 4 I practice, for calculatio of its, the defiitio usually is ot used. Istead, the kow results (the table of its) ad it properties ad theores are applied for calculatio of the it of the fuctios. The techique will deostrated i the followig eaples: ) (Liit of the cotiuous fuctio) Below, i the Sectio.4, we will see that for cotiuous fuctio f = f c. Therefore, we will eed just evaluatio of the fuctio at the poit c. The previous eaple is a siple cofiratio of this fact.

7 Chapter II CALCULUS II. Liits ad Cotiuity 6 But if the fuctio is cobiatio of differet fuctios, such as a ratioal fuctio or a product of trascedetal fuctios, calculatio ca yield ideteriates of the for:,,,,,,, which ca approach ay real uber or ifiity. The tricks to resolve these ideteriats will be show i eaples). 3) (L Hospital s Rule resolvig of the ideteriates or ) Fid si Direct calculatio of the fuctio at = yields the ideteriate. Marquis de l'hôpital ( 66-74) Fuctios f = si ad g = are defied ad differetiable o also the derivative g = o (, ). The accordig to Theore.7a: differetiate cotiuous fuctio si f cos = = = cos = g, ; 4) (Repeated applicatio of L Hospital s Rule Fid P Q ( ) where resolvig of the ideteriates or ) P = p p + p ad Q = q q + q are polyoials. Direct calculatio of the fuctio at yields the ideteriate ±. Fuctios f = P ad g = Q are differetiable o (, ) If <, the repeated differetiatio of the fuctios ties yields ( ) ( ) cotiuous fuctio for soe ( a, ) P P p = = = Q Q P If <, the repeated differetiatio of the fuctios ties yields ( ) ( ) cotiuous fuctio for soe ( a, ). P ad Q ( ) P ad Q ( ) P P R = = = ± = sg( q ) Q Q q If =, the repeated differetiatio of the fuctios ties yields ( ) ( ) cotiuous fuctio for soe ( a, ) P P p p = = = Q Q q q P ad Q ( )

8 6 Chapter II CALCULUS II. Liits ad Cotiuity 5) (Reductio to L Hospital s Rule resolvig of the ideteriates ) Fid l Direct calculatio of the fuctio at. = yields the ideteriate Trick: If the direct calculatio of the it f g rewritig the product of the fuctios as the ratio: yields, the f f g = g yields ideteriate, ad g f g = f yields ideteriate ad L Hospitals Rule ca be applied. I our case, let us do L' Hospital Rule cotiuous f l [ l ] l = = = = ( ) = 6) (Reductio to L Hospital s Rule resolvig of the ideteriates, etc) Fid a + Direct calculatio of the fuctio at = yields the ideteriate. Trick: If the direct calculatio of the it g epoetial ideteriate siilar to,, f, etc., the yields the rewrite the fuctio with the help of the idetity g g l f l a e = a, the g l f g l f c f = e = e = e a + a a a l + l + l + = e = e = e a a l l + + L'Hospital's Rule = e = e a a + a a + = e = e = e a

9 Chapter II CALCULUS II. Liits ad Cotiuity 63 7) (resolvig of the ideteriate ) Tricks: If the direct calculatio of the it f g yields the ideteriate, the oe of the followig tricks ca be used: a) rewrite the differece of the fuctios with the help of the idetities as a sigle fuctio (for eaple, su to product trigooetric idetities); b) ultiply ad divide by cojugate epressio: f g f + g f g = f + g c) rewrite as g f f g = f g Eaple: Fid + + Direct calculatio of the fuctio at yields the ideteriate. Multiply ad divide by cojugate epressio = = = = =

10 64 Chapter II CALCULUS II. Liits ad Cotiuity 8) (The Table of the Liits Rearkable Liits) k = k k = c si = cos = = = l = l = a + = e a a + a = e a. a. if < P p = = ± if > if Q q P = p p + p ad Q = q q + q 9. Raificatio: Defiitio of the Liit i the etric space. Let the fuctio f( ) :D be defied o the ope set D, where is a space with the etric d (,y) = y for all,y (etric = easure of the distace betwee poits ad y ). Cosider the eighborhood of the poit c called the ope ball: = { < < δ} ad Bδ ( c) B c V c δ (The Liit of a Fuctio etric space defiitio) We say that L is a it of the fuctio f whe approaches c f = L if for ay ε > there eists a δ > such that B c f B L L if the { } δ ε

11 Chapter II CALCULUS II. Liits ad Cotiuity CONTINUITY Fuctio is cotiuous if its graph is represeted by a cotiuous curve. For ore rigorous defiitio of the fuctio of a real variable, the topological cotiuity of the set of real ubers plays a sigificat role. The real lie which graphically represets all real ubers is cotiuous i the sese that it cotais o gaps: if we have two differet poits a b the o atter how close they are to each other there are still ifiitely ay poits betwee the (i fact their uber is equal to the uber of poits o the whole lie). O the other had, if we fi soe poit the there is o poit which is strictly et to it to the left or to the right. This topological structure of the real lie is its cotiuity. We just eted it to the graph of the fuctio by the rigorous atheatical defiitios. Fuctios which are used for odelig i egieerig or which describe soe physical properties or techological processes i ost cases are cotiuous by its ature. For eaple a teperature field i the ediu is cotiuous there are o sudde jups i its variatio. Chage of the teperature i tie at soe fied poit is also cotiuous for ay sudde chage of coditios. Eve ore obvious is cotiuity of the physical otio trajectories are cotiuous particles durig its trasitio i space attai all iterediate positios without ay gaps. But ituitive physical descriptios of cotiuity startig with the attepts ade by Aristotle i his Physics failed to be rigorous. Cotiuous fuctios posses soe rearkable properties iportat for the developet of the calculus, solutio of algebraic ad differetial equatios. DEFINITIONS: ) Cotiuity at a iterior poit of the doai Cosider the fuctio f ( ) defied o the doai D. Let the poit c D belogs to the doai D with soe ope iterval c a,b D f is defied at c with soe ope ( a,b ) : (eas that eighborhood). The the fuctio f ( ) is said to be cotiuous at the poit = c if f = f ( c) Note that i this defiitio i cotrast to the defiitio of the it the fuctio f has to be defied at = c. Therefore, the ε δ versio of the cotiuity at the poit is forulated as the followig f is said to be cotiuous at the poit = c if the fuctio if for ay ε > there eists a δ > such that < the if c δ f L < ε ) Cotiuity o the ope iterval The fuctio f ( ) is cotiuous o the ope iterval ( a,b) cotiuous at all poits of ( a,b ) : f = f ( ) for all ( a,b) 3) Cotiuity o the closed iterval The fuctio f ( ) is cotiuous o the closed iterval [ a,b] cotiuous i the ope iterval ( a,b ) ad f = f ( a) ad f = f ( b) + a b D if it is D if it is

12 66 Chapter II CALCULUS II. Liits ad Cotiuity 4) Piece-Wise Cotiuous Fuctio Let ck [ a,b], k =,,,..., with c = a ad c = b. The the fuctio f ( ) is said to be piece-wise cotiuous o the iterval [ a,b ] if f ( ) is cotiuous i all ope subitervals ( c k,c k + ), k =,,,..., (itervals of cotiuity). This fuctio looses its cotiuity at the isolated poits c [ a,b] call the poits of discotiuity. k which we ca 5) Poits of discotiuity are the eighborig poits of itervals of cotiuity f is ot defied. That eas that they are the poits at which the fuctio where the cotiuity of fuctios is broke. For eaple, for the fuctio f are the poits of discotiuity. =, the poits = π, =, ±, ±,... si f f Let fuctio f ( ) be defied ad cotiuous o the ope iterval of the poit ( a,b). The if there eist a it f = f a,b ecept the fuctio f ( ) ca be ade cotiuous o the whole iterval redefiig it i the followig way a,b by f f = f = I this case, the poit of discotiuity = is called reovable (see eaples i the Sectio). If there eist o-equal oe sided its of the fuctio f ( ) at its poit of discotiuity = f f f = f + the poit of discotiuity is called o-reovable. The value J = f f + f at f is ecessarily be defied at = ). The fuctio with the fiite uber of jups is called a piece-wise cotiuous fuctio. is called a jup of the fuctio + = (the fuctio

13 Chapter II CALCULUS II. Liits ad Cotiuity CONTINUITY THEOREMS: Theore. (properties of cotiuous fuctios - vector space C[ a,b ] ) g be cotiuous o the closed Let the fuctios f ad iterval [ a,b ]. The kf ( ) k f + g f g f (provided g ) g also are cotiuous fuctios o the iterval [ a,b ]. The set of all fuctios cotiuous o the iterval [ a,b ] is deoted by C[ a,b ] (it fors a vector space (see Chapter 3)). Theore. (coposite fuctio) Let the fuctios f ( ) be cotiuous o the closed iterval [ a,b ], ad let R = { f [ a,b] } be the rage of f ( ). Let the fuctio g be cotiuous o the set R. The the coposite fuctio y g f is cotiuous o [ a,b ]. = Theore.3 (etree value theore) Let the fuctio f ( ) be cotiuous o the closed iterval [ a,b ]. The f ( ) is bouded o [ a,b ]. This stateet ca be epaded to the followig: a) There eists c [ a,b] such that b) There eists d [ a,b] such that f c = i f [ a,b] f d = a f M [ a,b] Theore.4 (iterediate value theore Bolzao-Weierstrass Theore) Let the fuctio f ( ) be cotiuous o the closed iterval [ a,b ], ad let v be ay uber such that f ( a) < v< f ( b) if f ( a) < f ( b) or f ( b) < v< f ( a) if f ( a) > f ( b), the there eists a poit c [ a,b] such that f ( c) = v This theore eas that a cotiuous fuctio assues o the iterval of cotiuity all iterediate values.

14 68 Chapter II CALCULUS II. Liits ad Cotiuity This theore is the basis for the siplest ad ost popular ethod of fidig the roots of cotiuous fuctios: Bisectio Method If the fuctio f ( ) is cotiuous o [ a,b] ad f ( a) f ( b) < (it eas that ed values of the fuctio have the opposite sig, ad, therefore, is betwee f ( a ) ad f ( b ) ) the accordig the Iterediate Value Theore at soe poit c [ a,b] f ( c) = That eas that c is the root of equatio f = o [ a,b ]. Bisectio ethod is the uerical procedure of reductio of the iterval cotaiig the root by two at each step util the legth of the iterval becoes saller tha prescribed accuracy for the root. It cosists i the repetitio of the followig steps: let ε > be the prescribed accuracy, the ) if b a < ε the stop ) else a+ b c = 3) if f ( a) f ( c) < the b = c ad go to ) else a = c ad go to ) As a result, the value = c provides the root with the prescribed accuracy. See the Maple applicatio of the Bisectio Method. Cotiuity Raificatio: Defiitio 6) The fuctio f ( ) is uiforly cotiuous o ( a,b ) if for ay ε > there eists a δ > such that for all,y ( a,b) f f ( y) < ε such that y < δ. Uiforly cotiuous fuctio is also cotiuous. If fuctio is cotiuous o the closed iterval the it is also uiforly cotiuous.

15 Chapter II CALCULUS II. Liits ad Cotiuity REVIEW QUESTIONS: ) Give iforal ad foral defiitios of the it of the fuctios? ) What does it ea that it of the fuctio does ot eist? 3) What is the differece betwee the defiitios of the it of the fuctio at ifiity ad the ifiite it? 4) What is the cotiuity of the fuctio at the poit? 5) Give a eaple of the fuctio piece-wise cotiuous i. EXERCISES:. Show that iequality < c < δ is equivalet to two iequalities c δ < < c ad c< < c+ δ. Give ε δ defiitio of the it f = + 3. Usig defiitio of the it show that + = 7 d) a) 3 = 3 5 b) = e) + = 3 c) = f) = Fid the its a) h) b) + + i) + c) si 4 k) si d) ta l) si3 si5 cos e) 3 ) e f) 3+ ) ( )

16 7 Chapter II CALCULUS II. Liits ad Cotiuity g) + + o) si 5. Prove that the followig fuctio is cotiuous at the give poit: a) f = + at = 3 b) f = at = 6. Usig the Bisectio Method fid the root of the equatio i the idicated iterval with accuracy ε =. : a) b) 3 = o 3 + = o [, ]

17 Chapter II CALCULUS II. Liits ad Cotiuity 7 7. LIMITS AND CONTINUITY WITH MAPLE Calculatio of the its: > it(^+,=); 5 > it((-si())/cos()^,=pi/); > it(si()*l(),=); > it(sqrt(^+4*)-,=ifiity); > it(/,=); udefied > it(/^,=); Bisectio Method: Fid the root of the algebraic equatio 3+ = o the iterval [,4 ] with the accuracy ε =. > restart; > f():=^-3*+; f( ) := 3 + > plot(f(),=-..4); > a:=.;b:=4.;d:=b-a; > N:=5;:=; > epsilo:=.; a :=. b := 4. d := 3. N := 5 := ε :=.

18 7 Chapter II CALCULUS II. Liits ad Cotiuity > while (<N ad d>epsilo) do c:=(a+b)/; fac:=evalf(subs(=c,f()))*evalf(subs(=a,f())); if fac< the b:=c; else a:=c; ed if; d:=b-a; :=+ od; c :=.5 fac :=.5 d :=.5 := c := 3.5 fac := d :=.75 := 3 c :=.875 fac := d :=.375 := 4 c :=.6875 fac := d :=.875 := 5 c := fac := d :=.9375 Bisectio Method yields the result c = Eact root is = -.683