(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.
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1 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 approaches ifiity we mea the limit of f as moves icreasigly far to the right o the umber lie. Whe we say the limit of f as approaches egative ifiity ( ) we mea the limit of f as moves icreasigly far to the left. (The limit i each case may or may ot eist.) Lookig at f( ) (Figure.9), we observe (a) as, 0 ad we write lim 0, (b) as, 0 ad we write lim 0. We say that the lie y 0 is a horizotal asymptote of the graph of f. The graph of because f( ) has the sigle horizotal asymptote y lim ad lim. Lookig for Horizotal Asymptotes Use graphs ad tables to fid lim f ( ), lim f ( ), ad idetify all horizotal asymptotes of f(. Solve graphically ad cofirm umerically.
2 Figure.0 (a) The graph of has two horizotal asymptotes y = -ad y =. (b) Selected values of f. (Eample ) The Sadwich Theorem also holds for limits as Fidig a Limit as Approaches si Fid lim f( ) for f( ). Solve graphically ad umerically the cofirm aalytically.
3 Fid 5 si lim. Usig Theorem 5 a If the values of a fuctio f( ) outgrow all positive bouds as approaches a fiite umber a, we say that lim f ( ). If the values of f become large ad egative, eceedig all egative bouds a as a we say that lim f ( ). a Lookig at f( ) (Figure.9, p. 70), we observe that lim 0 ad lim. 0 We say that the lie 0 is a vertical asymptote of the graph of f. Fidig Vertical Asymptotes Fid the vertical asymptotes of f( ). the left ad right of each vertical asymptote. Describe the behavior to lim 0 lim 0 3
4 Fidig Vertical Asymptotes (si ) The graph of f ( ) ta has ifiitely may vertical (cos ) asymptotes, oe at each poit where the cosie is zero. If a is a odd multiple of, the lim ta ad lim ta, a as suggested by Figure. a You might thik that the graph of a quotiet always has a vertical asymptote where the deomiator is (si ) zero, but that eed ot be the case. For eample, we observed i Sectio. that lim. 0 Modelig Fuctios For Large Let f ( ) ad g( ) 3. Show graphically ad cofirm aalytically that while f ad g are quite differet for umerically small values of, they are virtually idetical for large. 4
5 If oe fuctio provides both a left ad right ed behavior model, it is simply called a ed behavior model. Thus, g( ) is a ed behavior model for 4 3 (Eample 6). f ( ) I geeral, g( ) a is a ed behavior model for the polyomial fuctio f ( ) a a a, a 0. Overall, the ed behavior of all polyomials behave like the ed behavior of moomials. This is the key to the ed behavior of ratioal fuctios, as illustrated i Eample 7. Fidig Ed Behavior Models Fid a ed behavior model for 5 4 (a) f( ) 3 57 (b) 3 g ( ) Fidig Ed Behavior Models Let f ( ) e. Show that g( ) is a right ed behavior model for f while h ( ) is a left ed behavior model for f. 5
6 We ca ivestigate the graph of y f ( ) as 0. as by ivestigatig the graph of y f Usig Substitutio Fid lim si. 6
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