MA Lesson 26 Notes Graphs of Rational Functions (Asymptotes) Limits at infinity

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1 MA 1910 Lesso 6 Notes Graphs of Ratioal Fuctios (Asymptotes) Limits at ifiity Defiitio of a Ratioal Fuctio: If P() ad Q() are both polyomial fuctios, Q() 0, the the fuctio f below is called a Ratioal Fuctio. P ( ) f( ) Q ( ) The followig are eamples of ratioal fuctios. 3 ( 3)( 3)( ) 3 f ( ), g( ), h( ), j( ) 3 3 ( 6) 1 If there are ay values of for which the deomiator of the fuctio would equal 0, these values must be ecluded from the domai of the fuctio. At those values of, the fuctio value does ot eist or is ot defied ad the graph of the fuctio is discotiuous at that value (has a break i the graph). Ay values of Q() for which Q() = 0, but P() 0, are values where there is a vertical asymptote. This is a vertical lie where the graph of f (fuctio values) approaches or as goes toward the umber we will call c (ay umber that makes the deomiator equal 0).. This is because lim f( ) or. Sice it is a vertical lie, its equatio is = c. c Fid the equatio(s) of ay vertical asymptotes for each ratioal fuctio. 1 E. 1: f( ) 3 E. : g ( ) 710 E 3: y E : j ( ) 3 1 E : 718 r ( ) ( 1)( 9) 1

2 Note: I eample of the previous page, there was a factor that cacelled out i both the umerator ad deomiator. That factor would ot give a vertical asymptote. IN FACT: THERE IS A HOLE IN THE GRAPH AT THE VALUE THAT MAKES THAT FACTOR BE ZERO. Fid the followig limits by dividig each term i the umerator ad deomiator by the highest power of i the deomiator. Also use the fact that the value of 1 approaches 0 as goes toward either or. (As a deomiator gets larger ad larger, but the umerator stays relative smaller; the value of the fractio is almost zero.) 1 1 lim 0 ad lim 0 This is the basic theorem for limits at ifiity. Fid the followig limits, if they eist. If a limit goes to or (gets larger without boud or smaller without boud), you may say the limit equals to or. E 6: E 7: lim lim lim a a 9a alim ( ) a a E 8: E 9: 3 3 lim 6 lim ( ) 3 6

3 I eamples 6 ad 7 of the previous page, the leadig term of the umerator had the same degree as the leadig term of the deomiator. The limit was the ratio of the coefficiets of those leadig terms. I eample 8 o the previous page, the leadig term of the umerator had a lesser degree tha the leadig term i the deomiator. The limit was zero. I eample 9 o the previous page, the leadig term of the umerator had a greater degree tha the leadig term i the deomiator. The limit was. **Fidig a horizotal asymptote of a ratioal fuctio (horizotal lie that the graph approaches as goes to or - ) uses the same procedure as the limits at ± i eamples 6 through 9. Here is a easy way to determie these limits or the equatio of ay horizotal asymptote.** 1. If the umerator has a lesser degree tha the deomiator, there is a horizotal asymptote with equatio y = 0 (the -ais).. If the umerator ad deomiator have the same degree, there is a horizotal asymptote with equatio y = the ratio of the leadig coefficiets. 3. If the umerator has a greater degree tha the deomiator, there is o horizotal asymptote. **Note: We will ot be discussig oblique asymptotes (lies other tha vertical or horizotal that a graph would approach; see eample 7 o page 781 of tet). Oe of the lies that the graph approaches is a slated lie. O ay assigmet problems i MyMathLab that ask for equatios of oblique asymptotes, aswer NO. You are ot resposible to fid ay oblique asymptotes o homework problems (paper or MyMathLab). E 10: Fid the equatios of ay horizotal or vertical asymptotes for these ratioal fuctios a) g( ) b) F( ) c) h( ) 9 3 3

4 Note: Equatios for horizotal asymptotes correspod to limits at ifiity or egative ifiity. Eve though we do ot usually describe a limit as (oly a umber), i MyMathLab, they may ecept as the aswer for a limit. E 11: Fid the followig limits. It a limit is ubouded, you may write or - for the limit value. 3 9 b 7 ( ) 3 a) lim ) lim 77 3 c) lim 3

5 Optioal: Sketch a graph of f usig asymptotes, itercepts, ad perhaps plottig a few poits. 6 3 f( )

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