1.3 LIMITS AT INFINITY; END BEHAVIOR OF A FUNCTION

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1 . Limits at Infinit; End Behavior of a Function 89. LIMITS AT INFINITY; END BEHAVIOR OF A FUNCTION Up to now we have been concerned with its that describe the behavior of a function f) as approaches some real number a. In this section we will be concerned with the behavior of f) as increases or decreases without bound. LIMITS AT INFINITY AND HORIZONTAL ASYMPTOTES If the values of a variable increase without bound, then we write, and if the values of decrease without bound, then we write. The behavior of a function f) as increases without bound or decreases without bound is sometimes called the end behavior of the function. For eample, 0 and 0 ) are illustrated numericall in Table.. and geometricall in Figure... 0 / / values Table.. 0, , conclusion As the value of / increases toward zero. As + the value of / decreases toward zero. In general, we will use the following notation. Figure Horizontal asmptote f) L.. its at infinit an informal view) If the values of f) eventuall get as close as we like to a number L as increases without bound, then we write f) L or f) L as ) Similarl, if the values of f) eventuall get as close as we like to a number L as decreases without bound, then we write f) L or f) L as ) L f) L + Horizontal asmptote Figure.. illustrates the end behavior of a function f when f) L or f) L In the first case the graph of f eventuall comes as close as we like to the line L as increases without bound, and in the second case it eventuall comes as close as we like to the line L as decreases without bound. If either it holds, we call the line L a horizontal asmptote for the graph of f. Figure.. f) L Eample It follows from ) and ) that 0 is a horizontal asmptote for the graph of f) / in both the positive and negative directions. This is consistent with the graph of / shown in Figure...

2 90 Chapter / Limits and Continuit 6 Eample Figure.. is the graph of f) tan. As suggested b this graph, tan π and tan π 6) ^ so the line π/ is a horizontal asmptote for f in the positive direction and the line π/ is a horizontal asmptote in the negative direction. Figure.. e tan 6 Figure.. + Eample graph, Figure.. is the graph of f) + /). As suggested b this + ) e and + ) e 7 8) so the line e is a horizontal asmptote for f in both the positive and negative directions. LIMIT LAWS FOR LIMITS AT INFINITY It can be shown that the it laws in Theorem.. carr over without change to its at + and. Moreover, it follows b the same argument used in Section. that if n is a positive integer, then ) n f))n f))n ) n f) 9 0) provided the indicated it of f) eists. It also follows that constants can be moved through the it smbols for its at infinit: kf) k f) kf) k f) ) provided the indicated it of f) eists. Finall, if f) k is a constant function, then the values of f do not change as or as,so k k k k ) Eample a) It follows from ), ), 9), and 0) that if n is a positive integer, then ) n ) 0 and n n 0 n b) It follows from 7) and the etension of Theorem..e) to the case that + ) [ + ) ] / [ + ) ] / e / e INFINITE LIMITS AT INFINITY Limits at infinit, like its at a real number a, can fail to eist for various reasons. One such possibilit is that the values of f) increase or decrease without bound as or as. We will use the following notation to describe this situation.

3 . Limits at Infinit; End Behavior of a Function 9.. infinite its at infinit an informal view) If the values of f) increase without bound as or as, then we write f) + or f) + as appropriate; and if the values of f) decrease without bound as or as, then we write as appropriate. f) or f) LIMITS OF n AS ± Figure.. illustrates the end behavior of the polnomials n for n,,, and. These are special cases of the following general results: {, n,,,... n +, n,,,... n +, n,, 6,... 6) Figure.. Multipling n b a positive real number does not affect its ) and 6), but multipling b a negative real number reverses the sign. Eample +, 76, 76 LIMITS OF POLYNOMIALS AS ± There is a useful principle about polnomials which, epressed informall, states: The end behavior of a polnomial matches the end behavior of its highest degree term.

4 9 Chapter / Limits and Continuit More precisel, if c n 0, then c0 + c + +c n n) c n n 7) c0 + c + +c n n) c n n 8) We can motivate these results b factoring out the highest power of from the polnomial and eamining the it of the factored epression. Thus, c 0 + c + +c n n n c 0 + c ) n + +c n n As or, it follows from Eample a) that all of the terms with positive powers of in the denominator approach 0, so 7) and 8) are certainl plausible. Eample ) ) 8 LIMITS OF RATIONAL FUNCTIONS AS ± One technique for determining the end behavior of a rational function is to divide each term in the numerator and denominator b the highest power of that occurs in the denominator, after which the iting behavior can be determined using results we have alread established. Here are some eamples. Eample 7 + Find 6 8. Solution. Divide each term in the numerator and denominator b the highest power of that occurs in the denominator, namel,. We obtain ) 6 8 ) Divide each term b. Limit of a quotient is the quotient of the its Limit of a sum is the sum of the its. A constant can be moved through a it smbol; Formulas ) and ).

5 . Limits at Infinit; End Behavior of a Function 9 Eample 8 Find a) + b) Solution a). Divide each term in the numerator and denominator b the highest power of that occurs in the denominator, namel,. We obtain Divide each term b. ) ) Limit of a quotient is the quotient of the its. Limit of a difference is the difference of the its A constant can be moved through a it smbol; Formula ) and Eample. Solution b). Divide each term in the numerator and denominator b the highest power of that occurs in the denominator, namel,. We obtain + + 9) In this case we cannot argue that the it of the quotient is the quotient of the its because the it of the numerator does not eist. However, we have 0, ) +, Thus, the numerator on the right side of 9) approaches + and the denominator has a finite negative it. We conclude from this that the quotient approaches ; that is, + + A QUICK METHOD FOR FINDING LIMITS OF RATIONAL FUNCTIONS AS + OR Since the end behavior of a polnomial matches the end behavior of its highest degree term, one can reasonabl conclude: The end behavior of a rational function matches the end behavior of the quotient of the highest degree term in the numerator divided b the highest degree term in the denominator.

6 9 Chapter / Limits and Continuit Eample 9 Use the preceding observation to compute the its in Eamples 7 and 8. Solution LIMITS INVOLVING RADICALS 0 ) ) Eample 0 Find a) + b) In both parts it would be helpful to manipulate the function so that the powers of are transformed to powers of /. This can be achieved in both cases b dividing the numerator and denominator b and using the fact that. Solution a). As, the values of under consideration are positive, so we can replace b where helpful. We obtain ) 6 ) + 0) 6 0) ) ) + ) ) 6 ) TECHNOLOGY MASTERY It follows from Eample 0 that the function f) + 6 has an asmptote of in the positive direction and an asmptote of in the negative direction. Confirm this using a graphing utilit. Solution b). As, the values of under consideration are negative, so we can replace b where helpful. We obtain ) + + 6

7 6 + Eample a) Find. Limits at Infinit; End Behavior of a Function ) b) 6 + ) Solution. Graphs of the functions f) 6 +, and g) 6 + for 0, are shown in Figure..6. From the graphs we might conjecture that the requested its are 0 and, respectivel. To confirm this, we treat each function as a fraction with a denominator of and rationalize the numerator. ) ) 6 + ) a) 6 + ) ) ) for >0 ) , 0 Figure..6 b) We noted in Section. that the standard rules of algebra do not appl to the smbols + and. Part b) of Eample illustrates this. The terms 6 + and both approach + as, but their difference does not approach 0. There is no it as + or. sin 6 + ) for >0 END BEHAVIOR OF TRIGONOMETRIC, EXPONENTIAL, AND LOGARITHMIC FUNCTIONS Consider the function f) sin that is graphed in Figure..7. For this function the its as and as fail to eist not because f) increases or decreases without bound, but rather because the values var between and without approaching some specific real number. In general, the trigonometric functions fail to have its as and as because of periodicit. There is no specific notation to denote this kind of behavior. In Section 0. we showed that the functions e and ln both increase without bound as Figures 0..8 and 0..9). Thus, in it notation we have Figure..7 ln + e + 0 ) For reference, we also list the following its, which are consistent with the graphs in Figure..8: e 0 ln ) 0 +

8 96 Chapter / Limits and Continuit e e e ln Figure..8 Figure..9 Finall, the following its can be deduced b noting that the graph of e is the reflection about the -ais of the graph of e Figure..9). e 0 e + ) QUICK CHECK EXERCISES. See page 00 for answers.). Find the its. a) ) b) ) ) c) ln d) e. Find the its that eist. a) b) c) + + sin + ). Given that f) and g) find the its that eist. a) [f) g)] b) c) d) f) g) f) + g) f) + g) 0 f)g). Consider the graphs of /, sin,ln, e, and e. Which of these graphs has a horizontal asmptote? EXERCISE SET. Graphing Utilit In these eercises, make reasonable assumptions about the end behavior of the indicated function.. For the function g graphed in the accompaning figure, find a) g) b) g). g). For the function φ graphed in the accompaning figure, find a) b) φ) φ). f) Figure E- Figure E-

9 . Limits at Infinit; End Behavior of a Function 97. For the function φ graphed in the accompaning figure, find a) φ) b) φ). f) b) Use Figure.. to find the eact value of the it in part a). 8. Complete the table and make a guess about the it indicated. f) / f) ,000 00,000,000,000 f) Figure E-. For the function G graphed in the accompaning figure, find a) G) b) G). G) Figure E-. Given that f), g), h) 0 find the its that eist. If the it does not eist, eplain wh. a) [f) + g)] b) [h) g) + ] c) e) g) [f)g)] d) [g)] + f) f ) g) h) + h) 6f) f) + g) 6. Given that f) 7 and g) 6 find the its that eist. If the it does not eist, eplain wh. a) c) e) g) [f) g)] b) [6f) + 7g)] [ + g)] d) [ g)] f)g) f ) [ f) + g) ] h) g) f) f) + )g) 7. a) Complete the table and make a guess about the it indicated. ) f) tan f) 0 + f) Find the its ) ) t t 7t s t t t s 7 s s ). ) e e.. + e + e. 7. ln e + e 6. e e ) ln + ) e + e e e ) + ) True False Determine whether the statement is true or false. Eplain our answer.. We have + ) + 0) + +.

10 98 Chapter / Limits and Continuit. If L is a horizontal asmptote for the curve f), then f) L and f) L. If L is a horizontal asmptote for the curve f), then it is possible for the graph of f to intersect the line L infinitel man times.. If a rational function p)/q)has a horizontal asmptote, then the degree of p) must equal the degree of q). FOCUS ON CONCEPTS. Assume that a particle is accelerated b a constant force. The two curves v nt) and v et) in the accompaning figure provide velocit versus time curves for the particle as predicted b classical phsics and b the special theor of relativit, respectivel. The parameter c represents the speed of light. Using the language of its, describe the differences in the long-term predictions of the two theories. Velocit c v v nt) Classical) Time v et) Relativit) t Figure E- 6. Let T ft) denote the temperature of a baked potato t minutes after it has been removed from a hot oven. The accompaning figure shows the temperature versus time curve for the potato, where r is the temperature of the room. a) What is the phsical significance of t 0 + ft)? b) What is the phsical significance of t + ft)? Temperature ºF) 7. Let Find a) 00 r T T ft ) Time min) +, < 0 f) + +, 0 t Figure E-6 f) b) f). 8. Let Find a) t + t t + 6, t <,000,000 gt) 6t 00, t >,000,000 t gt) b) t + gt). 9. Discuss the its of p) ) n as and for positive integer values of n. 0. In each part, find eamples of polnomials p) and q) that satisf the stated condition and such that p) + and q) + as. a) c) p) p) b) q) q) 0 p) + d) [p) q)] q). a) Do an of the trigonometric functions sin, cos, tan, cot, sec, and csc have horizontal asmptotes? b) Do an of the trigonometric functions have vertical asmptotes? Where?. Find c 0 + c + +c n n d 0 + d + +d m m where c n 0 and d m 0. [Hint: Your answer will depend on whether m<n, m n, orm>n.] FOCUS ON CONCEPTS These eercises develop some versions of the substitution principle, a useful tool for the evaluation of its.. a) Eplain wh we can evaluate e b making the substitution t and writing e t + et + b) Suppose g) + as. Given an function f), eplain wh we can evaluate f [g)] b substituting t g) and writing f [g)] ft) t + Here, equalit is interpreted to mean that either both its eist and are equal or that both its fail to eist.) c) Wh does the result in part b) remain valid if is replaced everwhere b one of, c, c, or c +?. a) Eplain wh we can evaluate e b making the substitution t and writing e t et 0 cont.)

11 . Limits at Infinit; End Behavior of a Function 99 b) Suppose g) as. Given an function f), eplain wh we can evaluate f [g)] b substituting t g) and writing f [g)] ft) t Here, equalit is interpreted to mean that either both its eist and are equal or that both its fail to eist.) c) Wh does the result in part b) remain valid if is replaced everwhere b one of, c, c, or c +? 6 Evaluate the it using an appropriate substitution.. 0 e/ e/ 7. 0 ecsc ecsc ln 9. [Hint: t ln ] ln 60. [ln ) ln + )] [Hint: t ] 6. ) [Hint: t ] 6. + ) [Hint: t /] 6. Let f) b, where 0 <b. Use the substitution principle to verif the asmptotic behavior of f that is illustrated in Figure 0... [Hint: f) b e ln b ) e ln b) ] 6. Prove that 0 + ) / e b completing parts a) and b). a) Use Equation 7) and the substitution t / to prove that ) / e. b) Use Equation 8) and the substitution t / to prove that 0 + ) / e. 6. Suppose that the speed v in ft/s) of a skdiver t seconds after leaping from a plane is given b the equation v 90 e 0.68t ). a) Graph v versus t. b) B evaluating an appropriate it, show that the graph of v versus t has a horizontal asmptote v c for an appropriate constant c. c) What is the phsical significance of the constant c in part b)? 66. The population p of the United States in millions) in ear t can be modeled b the function pt) +.e 0.0t 990) a) Based on this model, what was the U.S. population in 990? b) Plot p versus t for the 00-ear period from 90 to 0. c) B evaluating an appropriate it, show that the graph of p versus t has a horizontal asmptote p c for an appropriate constant c. d) What is the significance of the constant c in part c) for the population predicted b this model? 67. a) Compute the approimate) values of the terms in the sequence.0 0,.00 00, , , , What number do these terms appear to be approaching? b) Use Equation 7) to verif our answer in part a). c) Let a 9 denote a positive integer. What number is approached more and more closel b the terms in the following sequence?.0 a0a,.00 a00a,.000 a000a,.0000 a0000a, a00000a, a000000a... The powers are positive integers that begin and end with the digit a and have 0 s in the remaining positions). 68. Let f) + ). a) Prove the identit f ) f ) b) Use Equation 7) and the identit from part a) to prove Equation 8) The notion of an asmptote can be etended to include curves as well as lines. Specificall, we sa that curves f) and g) are asmptotic as + provided [f) g)] 0 and are asmptotic as provided [f) g)] 0 In these eercises, determine a simpler function g) such that f) is asmptotic to g) as or. Use a graphing utilit to generate the graphs of f) and g) and identif all vertical asmptotes. 69. f) [Hint: Divide into.] 70. f) + 7. f) f) + 7. f) sin + 7. Writing In some models for learning a skill e.g., juggling), it is assumed that the skill level for an individual increases with practice but cannot become arbitraril high. How do concepts of this section appl to such a model?

12 00 Chapter / Limits and Continuit 7. Writing In some population models it is assumed that a given ecological sstem possesses a carring capacit L. Populations greater than the carring capacit tend to decline toward L, while populations less than the carring capacit tend to increase toward L. Eplain wh these assumptions are reasonable, and discuss how the concepts of this section appl to such a model. QUICK CHECK ANSWERS.. a) + b) c) d) 0. a) b) does not eist c) e. a) 9 b). /, e, and e each has a horizontal asmptote. c) does not eist d). LIMITS DISCUSSED MORE RIGOROUSLY) In the previous sections of this chapter we focused on the discover of values of its, either b sampling selected -values or b appling it theorems that were stated without proof. Our main goal in this section is to define the notion of a it precisel, thereb making it possible to establish its with certaint and to prove theorems about them. This will also provide us with a deeper understanding of some of the more subtle properties of functions. MOTIVATION FOR THE DEFINITION OF A TWO-SIDED LIMIT The statement a f) L can be interpreted informall to mean that we can make the value of f) as close as we like to the real number L b making the value of sufficientl close to a. It is our goal to make the informal phrases as close as we like to L and sufficientl close to a mathematicall precise. To do this, consider the function f graphed in Figure..a for which f) L as a. For visual simplicit we have drawn the graph of f to be increasing on an open interval containing a, and we have intentionall placed a hole in the graph at a to emphasize that f need not be defined at a to have a it there. f) L f ) L + e L f) L + e f) L f) f) a L e L e 0 a 0 a Figure.. a) b) c) Net, let us choose an positive number ɛ and ask how close must be to a in order for the values of f) to be within ɛ units of L. We can answer this geometricall b drawing horizontal lines from the points L + ɛ and L ɛ on the -ais until the meet the curve f), and then drawing vertical lines from those points on the curve to the -ais Figure..b). As indicated in the figure, let 0 and be the points where those vertical lines intersect the -ais.