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1、HorizontalasymptoteTheHighSchoolAttachedtoNorthwestNormalUniversity王志强WangZhiqiang6thNovember2012Whenxapproachespositiveornegativeinfinity,thelimitsofrationalfunctionsasfollowing:、ReviewExample1:Example2:Ⅱ、Introducetothenewclass1.Observethefollowingfourfunctions,thendrawingt
2、hegraphsofthem::f(x)=1/x.gsp:f(x)=(1/x)+1.gsp:f(x)=x/x-1.gsp:f(x)=1/(x-1).gspExample4：Findthehorizontalasymptoteofthebelowfunction:Thehorizontalasymptotetellsus,roughly,wherethegraphwillgowhenxisreally,reallybig.SoWe'lllookatsomeverybigvaluesforx,somevaluesofxveryfarfromtheor
3、igin.Fromtheabovegraphs,weknowthelinesy=0andy=1arethehorizontalasymptotesofrationalfunctions.y=(x+2)/(x^2+1).gspAswecanseeinthetableofvaluesandthegraph,thehorizontalasymptoteisthex-axis.horizontalasymptote:y=0(thex-axis)Remark:Intheaboveexercise,thedegreeonthedenominator(name
4、ly,2)wasbiggerthanthedegreeonthenumerator(namely,1),andthehorizontalasymptotewasy=0(thex-axis).Thispropertyisalwaystrue.2.Whatisthedefinitionoftheasymptote?WhenacurveonavertexMalongthecurveinfiniteawayfromorigin,ifMtoastraightlinedistanceapproaches0,thenthislineiscalledasympt
5、oteofthecurve.当曲线上一点M沿曲线无限远离原点时，如果M到一条直线的距离无限趋近于0，那么这条直线称为这条曲线的渐近线。Example5：Findthehorizontalasymptoteofthefollowingfunction:Thisfunctionhasdegree-2polynomialstopandbottom;inparticular,thedegreesarethesameinthenumeratorandthedenominator.Question1:Whathappensifthedegreesarethe
6、sameinthenumeratoranddenominator?f(x)=(2x^2-11)/(x^+9).gspForbigvaluesofx,thegraphisveryclosetoy=2.So,wehaveTheliney=aisahorizontalasymptoteofthegraphofy=f(x)ifQuestion2:Whathappensifthedegreesarelargernumeratorthandenominator?Example6：Observethefollowingfunction,whatisthehor
7、izontalasymptoteofthisfunction?Thisfunctionhasdegree-3polynomialstopanddegree-2polynomialsbottom;thedegreesarelargernumeratorthandenominator.f(x)=(2x^3+4x^2-9)/(3-x^).gspFromtheabovegraph,weknowthisfunctionhasnohorizontalasymptoteandthelimitoff(x)asfollowing:Sothefunctionf(x)
8、hasnohorizontalasymptote.Example7：Observethefollowingfunction,whatis
