2.5. Infinite Limits and Vertical Asymptotes. Infinite Limits

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1 . Infinite Limits and Vertical Asmptotes. Infinite Limits and Vertical Asmptotes In this section we etend the concept of it to infinite its, which are not its as before, but rather an entirel new use of the term it. Infinite its provide useful smbols and language for describing the behavior of functions whose values become arbitraril large, positive or negative. We continue our analsis of graphs of rational functions from the last section, using vertical asmptotes and dominant terms for numericall large values of. You can get as low as ou want b taking close enough to. FIGURE.7 B You can get as high as ou want b taking close enough to. No matter how high B is, the graph goes higher. B No matter how low B is, the graph goes lower. One-sided infinite its: : + = q and : - = -q Infinite Limits Let us look again at the function ƒsd = >. As : +, the values of ƒ grow without bound, eventuall reaching and surpassing ever positive real number. That is, given an positive real number B, however large, the values of ƒ become larger still (Figure.7). Thus, ƒ has no it as : +. It is nevertheless convenient to describe the behavior of ƒ b saing that ƒ() approaches q as : +. We write ƒsd = + : : = q. + In writing this, we are not saing that the it eists. Nor are we saing that there is a real number q, for there is no such number. Rather, we are saing that : + s>d does not eist because > becomes arbitraril large and positive as : +. As : -, the values of ƒsd = > become arbitraril large and negative. Given an negative real number -B, the values of ƒ eventuall lie below -B. (See Figure.7.) We write ƒsd = - : : = -q. - Again, we are not saing that the it eists and equals the number - q. There is no real number - q. We are describing the behavior of a function whose it as : - does not eist because its values become arbitraril large and negative. EXAMPLE One-Sided Infinite Limits Find : + - and : - -. Geometric Solution The graph of = >s - d is the graph of = > shifted unit to the right (Figure.8). Therefore, = >s - d behaves near eactl the wa = > behaves near : FIGURE.8 Near =, the function = >s - d behaves the wa the function = > behaves near =. Its graph is the graph of = > shifted unit to the right (Eample ). : + - = q and : - - = - q. Analtic Solution Think about the number - and its reciprocal. As : +, we have s - d : + and >s - d : q. As : -, we have s - d : - and >s - d : - q.

2 6 Chapter : Limits and Continuit B No matter how high B is, the graph goes higher. EXAMPLE Discuss the behavior of ƒsd = near =, Two-Sided Infinite Limits f() gsd = near = -. s + d Solution As approaches zero from either side, the values of > are positive and become arbitraril large (Figure.9a): g() ( ) FIGURE.9 The graphs of the functions in Eample. ƒ() approaches infinit as :. g() approaches infinit as : -. The graph of gsd = >s + d is the graph of ƒsd = > shifted units to the left (Figure.9b). Therefore, g behaves near - eactl the wa ƒ behaves near. The function = > shows no consistent behavior as :. We have > : q if : +, but > : - q if : -. All we can sa about : s>d is that it does not eist. The function = > is different. Its values approach infinit as approaches zero from either side, so we can sa that : s> d = q. EXAMPLE (c) (d) (e) : : - : + - = - : - - = : s - d - Rational Functions Can Behave in Various Was Near Zeros of Their Denominators = : - - = : - - = : ƒsd = : : = q. gsd = : - : - s + d = q. s - d s - ds + d = : - s - ds + d = : - : + s - ds + d = - q - : - s - ds + d = q - s - ds + d + = does not eist. - + = The values are negative for 7, near. The values are positive for 6, near. See parts (c) and (d). - (f) : s - d = -s - d - : s - d = : s - d = - q In parts and the effect of the zero in the denominator at = is canceled because the numerator is zero there also. Thus a finite it eists. This is not true in part (f), where cancellation still leaves a zero in the denominator. Precise Definitions of Infinite Limits Instead of requiring ƒ() to lie arbitraril close to a finite number L for all sufficientl close to, the definitions of infinite its require ƒ() to lie arbitraril far from the ori-

3 . Infinite Limits and Vertical Asmptotes 7 f() gin. Ecept for this change, the language is identical with what we have seen before. Figures. and. accompan these definitions. DEFINITIONS Infinit, Negative Infinit as Limits B. We sa that ƒ() approaches infinit as approaches, and write ƒsd = q, : if for ever positive real number B there eists a corresponding d 7 such that for all 6 ƒ - ƒ 6 d Q ƒsd 7 B. FIGURE. :. ƒ() approaches infinit as. We sa that ƒ() approaches negative infinit as approaches, and write ƒsd = -q, : if for ever negative real number -B there eists a corresponding d 7 such that for all 6 ƒ - ƒ 6 d Q ƒsd 6 -B. The precise definitions of one-sided infinite its at eercises. are similar and are stated in the B EXAMPLE Using the Definition of Infinite Limits f() Prove that : = q. Solution Given B 7, we want to find d 7 such that FIGURE. ƒ() approaches negative infinit as :. Now, or, equivalentl, 6 ƒ - ƒ 6 d implies 7 B. 7 B if and onl if 6 B ƒ ƒ 6 B. Thus, choosing d = > B (or an smaller positive number), we see that ƒ ƒ 6 d implies 7 d Ú B. Therefore, b definition, : = q.

4 8 Chapter : Limits and Continuit Vertical asmptote Vertical Asmptotes Notice that the distance between a point on the graph of = > and the -ais approaches zero as the point moves verticall along the graph and awa from the origin (Figure.). This behavior occurs because asmptote asmptote, : + = q and : - = -q. We sa that the line = (the -ais) is a vertical asmptote of the graph of = >. Observe that the denominator is zero at = and the function is undefined there. Vertical asmptote, FIGURE. The coordinate aes are asmptotes of both branches of the hperbola = >. asmptote, Vertical asmptote, 6 FIGURE. The lines = and = - are asmptotes of the curve = s + d>s + d (Eample ). DEFINITION Vertical Asmptote A line = a is a vertical asmptote of the graph of a function = ƒsd if either ƒsd = ; q or ƒsd = ; q. :a + :a - EXAMPLE Looking for Asmptotes Find the horizontal and vertical asmptotes of the curve Solution We are interested in the behavior as : ; q and as : -, where the denominator is zero. The asmptotes are quickl revealed if we recast the rational function as a polnomial with a remainder, b dividing s + d into s + d. This result enables us to rewrite : We now see that the curve in question is the graph of = > shifted unit up and units left (Figure.). The asmptotes, instead of being the coordinate aes, are now the lines = and = -. EXAMPLE 6 = = + Asmptotes Need Not Be Two-Sided Find the horizontal and vertical asmptotes of the graph of ƒsd = Solution We are interested in the behavior as : ; q and as : ;, where the denominator is zero. Notice that ƒ is an even function of, so its graph is smmetric with respect to the -ais. The behavior as : ; q. Since :q ƒsd =, the line = is a horizontal asmptote of the graph to the right. B smmetr it is an asmptote to the left as well

5 . Infinite Limits and Vertical Asmptotes 9 Vertical asmptote, Vertical asmptote, asmptote, FIGURE. Graph of = -8>s - d. Notice that the curve approaches the -ais from onl one side. Asmptotes do not have to be two-sided (Eample 6). (Figure.). Notice that the curve approaches the -ais from onl the negative side (or from below). The behavior as : ;. Since the line = is a vertical asmptote both from the right and from the left. B smmetr, the same holds for the line = -. There are no other asmptotes because ƒ has a finite it at ever other point. EXAMPLE 7 The curves ƒsd = - q and ƒsd = q, : + : - Curves with Infinitel Man Asmptotes = sec = cos and = tan = cos sin both have vertical asmptotes at odd-integer multiples of p>, where cos = (Figure.). sec tan FIGURE. The graphs of sec and tan have infinitel man vertical asmptotes (Eample 7). The graphs of = csc = and = cot = cos sin sin have vertical asmptotes at integer multiples of p, where sin = (Figure.6). csc cot FIGURE.6 The graphs of csc and cot (Eample 7).

6 Chapter : Limits and Continuit EXAMPLE 8 A Rational Function with Degree of Numerator Greater than Degree of Denominator Find the asmptotes of the graph of 6 Vertical asmptote, The vertical distance between curve and line goes to zero as Oblique asmptote FIGURE.7 The graph of ƒsd = s - d>s - d has a vertical asmptote and an oblique asmptote (Eample 8). Solution We are interested in the behavior as : ; q and also as :, where the denominator is zero. We divide s - d into s - d: This tells us that ƒsd = ƒsd = - - = + + linear -. remainder Since : + ƒsd = q and : - ƒsd = -q, the line = is a two-sided vertical asmptote. As : ; q, the remainder approaches and ƒsd : s>d +. The line = s>d + is an oblique asmptote both to the right and to the left (Figure.7). Notice in Eample 8, that if the degree of the numerator in a rational function is greater than the degree of the denominator, then the it is + q or - q, depending on the signs assumed b the numerator and denominator as ƒ ƒ becomes large. Dominant Terms Of all the observations we can make quickl about the function in Eample 8, probabl the most useful is that This tells us immediatel that ƒsd = - - ƒsd = ƒsd L + ƒsd L - For numericall large For near If we want to know how ƒ behaves, this is the wa to find out. It behaves like = s>d + when is numericall large and the contribution of >s - d to the total

7 . Infinite Limits and Vertical Asmptotes value of ƒ is insignificant. It behaves like >s - d when is so close to that >s - d makes the dominant contribution. We sa that s>d + dominates when is numericall large, and we sa that >s - d dominates when is near. Dominant terms like these are the ke to predicting a function s behavior. Here s another eample. EXAMPLE 9 Two Graphs Appearing Identical on a Large Scale Let ƒsd = and gsd =. Show that although ƒ and g are quite different for numericall small values of, the are virtuall identical for ƒ ƒ ver large. Solution The graphs of ƒ and g behave quite differentl near the origin (Figure.8a), but appear as virtuall identical on a larger scale (Figure.8b).,, f() g(),, FIGURE.8 The graphs of ƒ and g, are distinct for ƒ ƒ small, and nearl identical for ƒ ƒ large (Eample 9). We can test that the term in ƒ, represented graphicall b g, dominates the polnomial ƒ for numericall large values of b eamining the ratio of the two functions as : ; q. We find that ƒsd : ;q gsd = so that ƒ and g are nearl identical for ƒ ƒ large : ;q = a - : ;q b =,