2.2 Limits Involving Infinity AP Calculus

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1 . Liits Ivolvig Iiity AP Calculus. LIMITS INVOLVING INFINITY We are goig to look at two kids o its ivolvig iiity. We are iterested i deteriig what happes to a uctio as approaches iiity (i both the positive ad egative directios), ad we are also iterested i studyig the behavior o a uctio that approaches iiity (i both the positive ad egative directios) as approaches a give value. Fiite Liits as ± Eaple: Ivestigate ad or =. 3 y Eaple: Ivestigate ad or =. y Both o these graphs have horizotal asyptotes. Are you able to deterie what they are ro the graphs? Deiitio: Horizotal Asyptote The lie y = b is a horizotal asyptote o the graph o a uctio y = () i either = or b = b I the last eaple, we saw that = 0. Usig this it ad the properties o its, we ca id the its o other uctios as approaches iiity. Eaple: 5 = 3

2 . Liits Ivolvig Iiity AP Calculus Eaple: Fid each o the its. Deterie whether or ot there are ay horizotal asyptotes. I so, where? a) b) c) 3 5 d) 3 5 e) I all o the eaples above, i there was a it as approached positive iiity, the it as approached egative iiity was the sae. Thus, there was oe horizotal asyptote, ad the uctio approached this sae asyptote i both directios. However, certai uctios, especially irratioal uctios, have ore tha oe horizotal asyptote. Eaple: Ivestigate 3 ad 3 3

3 . Liits Ivolvig Iiity AP Calculus Ed Behavior Models Deiitio: Ed Behavior Model For a ratioal uctio a b behavior odel ca be writte as, where is the degree o the uerator ad is the degree o the deoiator. The ed a b or a. b We ca use the ed behavior odels o ratioal uctios to idetiy ay horizotal asyptotes the uctio. Eaple: Go back ad look at the last si eaples. Fid the ed behavior odels or each. For ratioal uctios we have the ollowig results. I = a ad g b dieret ors. = the g takes o three = Ed Behavior Model Ed Behavior Asyptote < > Eaple: I the last sectio we proved that si =. Ivestigate 0 si. 33

4 . Liits Ivolvig Iiity AP Calculus It should coe as o surprise that the proo o this it ivolves the Sadwich Theore. See the tet i you just ca't cotai your curiosity. The it properties that we used whe approaches c are still valid as approaches iiity. Eaple: 5 si si Eaple: Iiite Liits as a A secod type o it ivolvig iiity is to deterie the behavior o the uctio as approaches a certai value whe the uctio icreases or decreases without boud. Eaple: Ivestigate 0 ad. 0 I the last eaple we call the lie = 0 a vertical asyptote. Deiitio: Vertical Asyptote The lie = a is a vertical asyptote o the graph o a uctio y = () i either a = ± or a = ± Iportat : Iiity is NOT a uber, ad thus the it FAILS to eist i both o these cases. I this sees cousig, the use the otatio as a (ro the right or let), the the uctio () ±. Eaple: Fid the vertical asyptotes o (). Describe the behavior o () to the let ad right o each asyptote. a) ( ) = 4 b) ( ) = 53 c) ( ) =

5 . Liits Ivolvig Iiity AP Calculus Eaple: Sketch the uctio that satisies the stated coditios. = 5 5 = = 0 = = Eaple: Sketch the uctio that satisies the stated coditios. = 4 4 = = Eaple: Aswer the ollowig questios: a) How do you id horizotal asyptotes? b) How do you id vertical asyptotes? c) How do you id oblique (slated) asyptotes? Notecards ro Sectio.: Deiitio o a horizotal asyptote, Deiitio o a vertical asyptote, Ed Behavior, Ed Behavior Models, Oblique (Slat) Asypototes 35

(Figure 2.9), we observe x. and we write. (b) as x, x 1. and we write. We say that the line y 0 is a horizontal asymptote of the graph of f.

(Figure 2.9), we observe x. and we write. (b) as x, x 1. and we write. We say that the line y 0 is a horizontal asymptote of the graph of f. The symbol for ifiity ( ) does ot represet a real umber. We use to describe the behavior of a fuctio whe the values i its domai or rage outgrow all fiite bouds. For eample, whe we say the limit of f as