Limit of a Function

1 Limit of a FunctionChapter 2In This ChapterMany topics are included in a typical course in calculus. But the three most fun-damental topics in this study are the concepts of Limit , derivative, and integral. Each of these con-cepts deals with functions , which is why we began this text by first reviewing some importantfacts about functions and their , two problems are used to introduce the basic tenets of calculus. These are thetangent line problemand the area problem. We will see in this and the subsequent chapters thatthe solutions to both problems involve the Limit An Informal That Involve A Formal Tangent Line ProblemChapter 2 in Reviewy (x)Lax a xy (x) L (x) Lx a 9/26/09 5:20 PM Page 67 Jones and Bartlett Publishers, LLC.

2 NOT FOR SALE OR DISTRIBUTION. 68 CHAPTER 2 Limit of a An Informal ApproachIntroductionThe two broad areas of calculus known as differentialand integral calculusare built on the foundation concept of a Limit . In this section our approach to this important con-cept will be intuitive, concentrating on understanding whata Limit is using numerical andgraphical examples. In the next section, our approach will be analytical, that is, we will use al-gebraic methods to computethe value of a Limit of a of a Function Informal ApproachConsider the Function (1)whose domain is the set of all real numbers except . Although fcannot be evaluated atbecause substituting for xresults in the undefined quantity 0 0,can be calcu-lated at any number xthat is very closeto . The two tables 4f (x) 4 4 4f (x) 16 x24 xshow that as x approaches from either the left or right, the Function values appearto be approaching 8, in other words, when xis near is near 8.

3 To interpret the numer-ical information in (1) graphically, observe that for every number , the Function f canbe simplified by cancellation:As seen in FIGURE , the graph of fis essentially the graph of with the excep-tion that the graph of fhas a holeat the point that corresponds to . For xsufficientlyclose to , represented by the two arrowheads on the x-axis, the two arrowheads on the y-axis,representing Function values , simultaneously get closer and closer to the number , in view of the numerical results in (2), the arrowheads can be made as close as weliketo the number 8. We say 8 is the limitof as x approaches .Informal DefinitionSuppose L denotes a finite number. The notion of approaching L asx approaches a number a can be defined informally in the following manner.

4 If can be made arbitrarily close to the number L by taking x sufficiently closeto but different from the number a, from both the left and right sides of a, then thelimitofas x approaches a is discussion of the Limit concept is facilitated by using a special notation. If welet the arrow symbolrepresent the word approach, then the symbolismindicates that x approaches a number a from the left,that is, through numbers that are less than a, andsignifies that x approaches a from the right,that is, through numbers that are greater than a. Finally, the notationsignifies that x approaches a from both sides,in other words, from the left and the right sides of aon a number line. In the left-hand tablein (2) we are letting (for example,is to the left of on the number line),whereas in the right-hand table.

5 One-Sided LimitsIn general, if a Function can be made arbitrarily close to a number L1by taking xsufficiently close to, but not equal to, a number afrom the left, then we write(3)f (x)SL1 as xSa or limxSa f (x) (x)xS 4 4 4 xSaxSa xSa Sf (x)f (x)f (x) 4f (x)f (x) 4x 4y 4 xf (x) 16 x24 x (4 x)(4 x)4 x 4 4 4, f (x)f (x) 4x (x) (x) (2)xy 4y816 x24 xFIGURE xis near ,is near 8f (x) 9/26/09 5:20 PM Page 68 Jones and Bartlett Publishers, LLC. NOT FOR SALE OR DISTRIBUTION. The number L1is said to be the left-hand Limit ofas xapproaches , if canbe made arbitrarily close to a number L2by taking xsufficiently close to, but not equal to, a num-ber afrom the right, then L2is the right-hand Limit ofas xapproaches aand we write(4)The quantities in (3) and (4) are also referred to as one-sided LimitsIf both the left-hand Limit and the right-hand Limit exist and have a common value L,then we say that Lis the Limit of as xapproaches aand write(5)A Limit such as (5) is said to be a two-sided Limit .

6 See FIGURE Since the numerical tablesin (2) suggest that(6)we can replace the two symbolic statements in (6) by the statement(7)Existence and NonexistenceOf course a Limit (one-sided or two-sided) does not have toexist. But it is important that you keep firmly in mind: The existence of a Limit of a Function f as x approaches a(from one side or fromboth sides),does not depend on whether f is defined at a but only on whether f isdefined for x near the number example, if the Function in (1) is modified in the following mannerthen is defined and but still See FIGURE In general,the two-sided Limit does not exist if either of the one-sided limits or fails to exist, or if and but EXAMPLE 1A Limit That ExistsThe graph of the Function is shown in FIGURE As seen from thegraph and the accompanying tables, it seems plausible thatand consequently limxS4 f (x) f (x) 6 and limxS4 f (x) 6f (x) x2 2x 2L1 f (x)

7 L2,limxSa f (x) L1limxSa f (x)limxSa f (x)limxSa f (x)limxS 4 16 x24 x ( 4) 5,f ( 4)f (x) 16 x24 x,x 45,x 4,f (x)S8 as xS 4 or equivalently limxS 4 16 x24 x (x)S8 as xS 4 and f (x)S8 as xS 4 ,limxSa f (x) (x)limxSa f (x) L and limxSa f (x) L,limxSa f (x)limxSa f (x)f (x)SL2 as xSa or limxSa f (x) (x)f (x)f (x) limits An Informal Approach69xS4 (x) (x) that in Example 1 the given Function is certainly defined at 4, but at no time didwe substitute into the Function to find the value of limxS4 f (x).x 4y (x)Lax a xy (x) L (x) Lx a x 4y8y 16 x24 x,5,x 4x 4y x2 2x 2xy 64 FIGURE if andonly if as andas xSa f (x)SLxSa f (x)SLxSaf (x)SLFIGURE fis defined at aoris not defined at ahas no bearing on theexistence of the Limit of as xSaf (x)FIGURE of Function inExample 9/26/09 5:20 PM Page 69 Jones and Bartlett Publishers, LLC.

8 NOT FOR SALE OR DISTRIBUTION. EXAMPLE 2A Limit That ExistsThe graph of the piecewise-defined functionis given in FIGURE Notice that is not defined, but that is of no consequence whenconsidering From the graph and the accompanying tables,limxS2 f (x).f (2)f (x) ex2,x62 x 6,x7270 CHAPTER 2 Limit of a FunctionxS2 (x) (x) see that when we make x close to 2, we can make arbitrarily close to 4, and soThat is,EXAMPLE 3A Limit That Does Not ExistThe graph of the piecewise-defined functionis given in FIGURE From the graph and the accompanying tables, it appears that as xapproaches 5 through numbers less than 5 that Then as xapproaches 5 through numbers greater than 5 it appears that But sincewe conclude that does not f (x)limxS5 f (x) limxS5 f (x),limxS5 f (x) f (x) (x) ex 2,x 5 x 10,x75lim xS2f (x) f (x) 4 and limxS2 f (x) (x)xS5 (x) (x)

9 4A Limit That Does Not ExistRecall, the greatest integer functionorfloor functionis defined to be the greatestinteger that is less than or equal to x. The domain of fis the set of real numbers . Fromthe graph in FIGURE see that is defined for every integer n; nonetheless, for eachinteger n,does not exist. For example, as xapproaches, say, the number 3, the two one-sided limits exist but have different values:(8)In general, for an integer n,EXAMPLE 5A Right-Hand LimitFrom FIGURE should be clear that as that isIt would be incorrect to write since this notation carries with it the connotationthat the limits from the left and from the right exist and are equal to 0. In this case does not exist since is not defined for (x) 1xlimxS0 1xlimxS01x 0limxS0 1x ,f (x) 1xS0limxSn f (x) n 1 whereas limxSn f (x) f (x) 2 whereas limxS3 f (x) f (x)f (n)( q, q)f (x) :x;xy42 FIGURE of Function inExample 2 FIGURE of Function inExample 3xy575xy11 12345234 2y x xxyy xFIGURE of Function inExample 4 FIGURE of Function inExample 5 The greatest integer Function wasdiscussed in Section 9/26/09 5:20 PM Page 70 Jones and Bartlett Publishers, LLC.

10 NOT FOR SALE OR DISTRIBUTION. If is a vertical asymptote for the graph of then will always failto exist because the Function values must become unbounded from at least one side ofthe line EXAMPLE 6A Limit That Does Not ExistA vertical asymptote always corresponds to an infinite break in the graph of a Function f. InFIGURE see that the y-axis or is a vertical asymptote for the graph of The tablesf (x) 1> 0x (x)limxSa f (x)y f (x),x limits An Informal Approach71xS0 (x)101001000xS0 (x) 10 100 1000clearly show that the Function values become unbounded in absolute value as we getclose to 0. In other words,is not approaching a real number as nor as Therefore, neither the left-hand nor the right-hand Limit exists as xapproaches 0.