Salas, Hille, Etgen Calculus: One and Several VariablesCopyright 2007 ohn !iley " Sons, #nc$%ll rights reserved$Main MenuSection 2$& 'he (i)it *rocess +%n #ntuitive #ntroduction,a$ 'he (i)it *rocessb$ %rea o- a .egion /ounded by a Curvec$ 'he #dea o- a (i)itd$ E0a)plee$ #llustration o- a (i)it-$ (i)its on Various 1unctionsg$ E0a)pleh$ One2Sided (i)itsi$ E0a)ple: One2Sided (i)its3$ %nother E0a)ple4$ 1unctions %pproaching #n-inityl$ Su))ary o- (i)its that 5o 6ot E0istSection 2$2 5e-inition o- (i)ita$ 5e-initionb$ #llustration o- 5e-initionc$ Selection o- Epsilond$ (i)its on Open #ntervalse$ (i)it *roperties-$ E7uivalent (i)it *ropertiesg$ (e-t2hand and .ight2hand (i)itsh$ E0a)pleSection 2$8 So)e (i)it 'heore)sa$ 'he 9ni7ueness o- a (i)itb$ (i)it *roperties: %rith)etic o- (i)itsc$ (i)it o- :uotientsd$ (i)its that 5o 6ot E0ist -or :uotientsChapter 2: (i)its and ContinuitySection 2$; Continuitya$ Continuity at a *ointb$ 'ypes o- 5iscontinuityc$ *roperties o- Continuityd$ E0a)plee$ Co)position 'heore)-$ One2sided Continuityg$ Continuity on #ntervalsSection 2$< 'he *inching 'heore)= 'rigono)etric (i)itsa$ 'he *inching 'heore)b$ /asic 'rigono)etric (i)itsc$ Continuity o- the 'rigono)etric (i)itsd$ E0a)ple Section 2$> '?o /asic 'heore)sa$ 'he #nter)ediate2Value 'heore)b$ /oundedness= E0tre)e Valuesc$ 'he E0tre)e2Value 'heore)d$ *roperties o- the '?o /asic 'heore)sSalas, Hille, Etgen Calculus: One and Several VariablesCopyright 2007 ohn !iley " Sons, #nc$%ll rights reserved$Main Menu'he (i)it *rocessTHE LIMIT PROCESS (AN INTUITIVE INTRODUCTION)!e could begin by saying that li)its are i)portant in calculus, but that ?ould be a )a3or understate)ent$ Without limits, calculus would not exist. Every single notion of calculus is a limit in one sense or another$ 1or e0a)ple:!hat is the slope o- a curve@ #t is the li)it o- slopes o- secant lines$ !hat is the length o- a curve@ #t is the li)it o- the lengths o- polygonal paths inscribed in the curve$ Salas, Hille, Etgen Calculus: One and Several VariablesCopyright 2007 ohn !iley " Sons, #nc$%ll rights reserved$Main Menu'he (i)it *rocess!hat is the area o- a region bounded by a curve@ #t is the li)it o- the su) o- areas o- appro0i)ating rectangles$Salas, Hille, Etgen Calculus: One and Several VariablesCopyright 2007 ohn !iley " Sons, #nc$%ll rights reserved$Main Menu'he (i)it *rocessThe Idea of a Limit!e start ?ith a nu)ber c and a -unction f de-ined at all nu)bers x near c but not necessarily at c itsel-$ #n any case, ?hether or not f is de-ined at c and, i- so, ho? is totally irrelevant$6o? let L be so)e real nu)ber$ !e say that the limit off +x, as x tends to c is L and ?riteprovided that +roughly spea4ing,as x approaches c, f(x) approaches Lor +so)e?hat )ore precisely, provided thatf +x, is close to L for all x A c which are close to c.( )li)x cf x LSalas, Hille, Etgen Calculus: One and Several VariablesCopyright 2007 ohn !iley " Sons, #nc$%ll rights reserved$Main Menu'he (i)it *rocessExampleSet f+x, B;x C < and ta4ec B 2$ %s 0 approaches 2, ;x approaches D and ;x C < approaches D C < B &8$ !e conclude that$ &8 , + li)2x fxSalas, Hille, Etgen Calculus: One and Several VariablesCopyright 2007 ohn !iley " Sons, #nc$%ll rights reserved$Main Menu'he (i)it *rocessExample ! Set%s x approaches ED, & E x approaches F and approaches 8$ !e conclude that#- -or that sa)e -unction ?e try to calculate?e run into a proble)$ 'he -unctionis de-ined only -or x G &$ #t is there-ore not de-ined -or x near 2, and the idea o- ta4ing the li)it as x approaches 2 )a4es no sense at all:does not e0ist.( )& f x x and ta4e c B ED$& x ( )Dli) 8xf x( )2li)xf x( )& f x x ( )2li)xf xSalas, Hille, Etgen Calculus: One and Several VariablesCopyright 2007 ohn !iley " Sons, #nc$%ll rights reserved$Main Menu'he (i)it *rocessExample " 1irst ?e ?or4 the nu)erator: as x approaches 8, x8 approaches 27, E2x approaches H>, and x8 H 2x C ; approaches 27 H > C ; B 2$ !e conclude that( )28 8Fli) li) 8 >8x xxxx + Salas, Hille, Etgen Calculus: One and Several VariablesCopyright 2007 ohn !iley " Sons, #nc$%ll rights reserved$Main Menu'he (i)it *rocessExample < ?ould -orce the li)it to be ;$or a full limit to exist, !oth one"sided limits have to exist and they have to !e e#ual.( )x H 8+x2 H 8x H ;,2 x H ;x C & +2x2 C 7x C x x x xx x + + +Salas, Hille, Etgen Calculus: One and Several VariablesCopyright 2007 ohn !iley " Sons, #nc$%ll rights reserved$Main MenuContinuitySalas, Hille, Etgen Calculus: One and Several VariablesCopyright 2007 ohn !iley " Sons, #nc$%ll rights reserved$Main MenuContinuityExample !'he -unction +x, Bis continuous at all nu)bers greater than 8$ 'o see this, note thatB f g, ?hereandg+x, B $ 6o?, ta4e any c 8$ Since g is a rational -unction and g is de-ined at c, g is continuous at c$ %lso, since g+c, is positive and - is continuous at each positive nu)ber, f is continuous at g+c,$ /y 'heore) 2$;$;,is continuous at c$x2 C &x H 8x2 C &x H 8x x f , +Salas, Hille, Etgen Calculus: One and Several VariablesCopyright 2007 ohn !iley " Sons, #nc$%ll rights reserved$Main MenuContinuityExample "'he -unctionis continuous every?here e0cept at 0 B W8, ?here it is not de-ined$ 'o see this, note thatB fg'h, ?here and observe that each o- these -unctions is being evaluated only ?here it is continuous$ #n particular, g and h are continuous every?here, f is being evaluated only at nonJero nu)bers, and ' is being evaluated only at positive nu)bers$ $ &> , + , , + , < , + ,&, +2+ x x h x x ' x x gxx fSalas, Hille, Etgen Calculus: One and Several VariablesCopyright 2007 ohn !iley " Sons, #nc$%ll rights reserved$Main MenuContinuitySalas, Hille, Etgen Calculus: One and Several VariablesCopyright 2007 ohn !iley " Sons, #nc$%ll rights reserved$Main MenuContinuityExample /5eter)ine the discontinuities, i- any, o- the -ollo?ing -unction:f+x, B 2x C &,x 0 &, 0 x & x2 C &, x &$ +1igure 2$;$D,Salas, Hille, Etgen Calculus: One and Several VariablesCopyright 2007 ohn !iley " Sons, #nc$%ll rights reserved$Main MenuContinuityExample 05eter)ine the discontinuities, i- any, o- the -ollo?ing -unction: f+x, B x8,x H& x2 H 2, H& x & > H x, &x ;,; x 7

7 H xSalas, Hille, Etgen Calculus: One and Several VariablesCopyright 2007 ohn !iley " Sons, #nc$%ll rights reserved$Main MenuContinuity( )2&&f xxCo#ti#,it- o# I#te)4al%% -unction f is said to be continuous on an interval i- it is continuous at each interiorpoint o- the interval and one2sidedly continuous at ?hatever endpoints the interval )aycontain$1or e0a)ple:(i) 'he -unction is continuous on ME&, &N because it is continuous at each point o- +E&, &,, continuous -ro) the right at E&, and continuous -ro) the le-t at &$ 'he graph o- the -unction is the se)icircle$(ii)'he -unction is continuous on +E&, &, because it is continuous at each point o- +E&, &,$ #t is not continuous on ME&, &, because it is not continuous -ro) the right at E&$ #t is not continuous on +E&, &N because it is not continuous -ro) the le-t at &$(iii) 'he -unction graphed in 1igure 2$;$D is continuous on +EO, &N and continuous on +&,O,$ #t is not continuous on M&,O, because it is not continuous -ro) the right at &$(i4) *olyno)ials, being every?here continuous, are continuous on +EO,O,$Continuous -unctions have special properties not shared by other -unctions$( )2& f x x Salas, Hille, Etgen Calculus: One and Several VariablesCopyright 2007 ohn !iley " Sons, #nc$%ll rights reserved$Main Menu'rigono)etric (i)itsSalas, Hille, Etgen Calculus: One and Several VariablesCopyright 2007 ohn !iley " Sons, #nc$%ll rights reserved$Main Menu'rigono)etric (i)its1ro) this it -ollo?s readily thatSalas, Hille, Etgen Calculus: One and Several VariablesCopyright 2007 ohn !iley " Sons, #nc$%ll rights reserved$Main Menu'rigono)etric (i)itsSalas, Hille, Etgen Calculus: One and Several VariablesCopyright 2007 ohn !iley " Sons, #nc$%ll rights reserved$Main Menu'rigono)etric (i)its#n )ore general ter)s,Example 1indSol,tio#'o calculate the -irst li)it, ?e Rpair o--S sin ;x ?ith ;x and use +2$,:'here-ore,'he second li)it can be obtained the sa)e ?ay:0 0sin ; & cos 2li) and li)8