Exercises-If the limit ofdenominatoris zero,limits ofFind the limits in Exercises 11-22.functions can befound by creating and cancelling a12. lim(y2 + 5x 2)11. lim (2x + 5)commonfactor.13. Jim 8(t 5)( 7)14. im,( 2r* + 4r + 8)V2+h-J2Example4:Evaluatelim15.hh-t16, 3(2 - 1)x+ 6y+ 217. lim, 3(2x 1)218.jp+5y+619. lim,(5-y/20. lim (22 8)/V5h + 4-222.lim21. JmV3h +1 +1AhExercisesExercisesLimits of quotieats Find the limits in Exercises 23-42.V-34x35.lim36. limx + 349x-924jim7 + 4x +3N2-VxV2+8-325m102 7x + 1037. lim38. lim26. Jmx+5x+1X - 2+1Vx+3-272 +1 2r2 + 3t + 2V+12- 427. 1m+.228. linm39.lim40.jmV+5-37-117-1-2x 225y3 + 8y230.jm.3y-16y2-V2-54-41.lim42.limx+3-45-Vx2+9-1古31.32, limT-1.833#号34. lim2416Limits of some functions can be found byExercisesSandwichTheorem.~an.indirectwayLimits with trigonometric functions Find the limits in Exercises4350.44. lim sin2x43. lim (2 sinx 1)45. lim sec.x46. lim tan x+x+sinx47. lim48. lim (2 1)(2 cos x)3cosx49. lim Vx + 4 cos (x + m) 50. lim V7 + sec x0FIGURE2.12Thegraph of f issandwiched between the graphs of g and h
2016/11/15 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley • If the limit of denominator is zero, limits of functions can be found by creating and cancelling a common factor. Example 4: Evaluate 0 2 2 lim h h h Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Exercises Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Exercises Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Exercises Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Exercises Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley • Limits of some functions can be found by Sandwich Theorem. ~ an indirect way
THEOREM4The Sandwich TheoremTHEOREMSIf f(x) g(x) for all x in some open interval containing c, exceptSuppose that g(x) s f(x) s h(x) for all x in some open interal containing c,possibly at x = c. itself, and the limits of f and g both exist as x approaches cexcept possibly at x c itself, Suppose also thatthenin g(c) - lin Mx) - L.lim f(x) s lim g(x)Then lim, f(x) L.EXAMPLE10Given that≤)≤1+号1-forallx *o.find lim, u(r), no matter how complicated ir is.=-10l(b)(a)FIGURE 2.14The Sandwich Theorem showsthat (a) limeo sin e 0 and(b) lime-o (1 -- cos 0) - 0 (Example 11).FIGURE 2:13Anyfunction (x) whosegraph lies in the region betweny = 1 + (x*/2) and y = 1 - (x /4) haslimit 1as x→0 (Example 10).Thefunctionhaslimity=:x1asxapproaches0from the right, andlimit -1from theleft.3.One-Sided Limitslim f(x)=1 andlim f(x)=-1x0X-→0FIGURE2.24Different right-hand andleff-hand limits at the origin
2016/11/15 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3. One-Sided Limits Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The function has limit 1 as x approaches 0 from the right, and limit -1 from the left. 0 0 lim ( ) 1 lim ( ) 1 x x f x and f x
DefinitionRight-Hand and Leff-Hand LimitsLetf(x) be defined on an interval (c,a), where d<a.MIf f(x) approaches arbitrarily close to M/as xapproachesa from within the interval,thenwe saythatfhas left-hand limttMata,(b)m-Mandwewriterirnf(x)=M(b)im-MFIGURE2.25(a)Righ-handlimitasapproacbesc (b)Lf-handlimitas(a)im-1approachFIGURE2.25(a) Right-hand limit as x approacbes e. (b) Leff-hand limit as xLet f(x) be defined on an interval (a,b), where a<b.approaches c.Iff(x)approaches arbitrarilycloseto Las xapproaches a from within that interval, thenwe say that f has right-hand limit L ataandwewritelim(x)=LV4-rTHEOREM6Afunction f(oy has alimit asx approaches cif and only ifithas lef-hand andright-hand limits there and these one-sided limits are equalm Fu) = Llim, f(x) = Llim (x) = L.eandlimV4-x2-0andFIGURE2.26lim,V4-2=0(Example1)ExercisesFinding Limits Graphically1. Which of the following statemenits about the function y f(x)graphed here are truc, and which are false?-fUxs0im, /Gx) 1alim_ f(x) = 0eFIGURE2.27Graph of the functionlimn f(x)existsfjimf(x)=0in Example 2.lim)=1lim(x)=1gi.lim yx)-o()-2f(x) dolim, (x) = 0
2016/11/15 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Definition Right-Hand and Left-Hand Limits Let f(x) be defined on an interval (a,b), where a<b. If f(x) approaches arbitrarily close to L as x approaches a from within that interval, then we say that f has right-hand limit L at a, and we write lim ( ) x a f x L Let f(x) be defined on an interval (c,a), where c<a. If f(x) approaches arbitrarily close to M as x approaches a from within the interval, then we say that f has left-hand limit M at a, and we write lim ( ) x a f x M Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Exercises