58.BusinessThe cost function for a new supermarket to recycle tons of organic material is given bywhere is the cost (in dollars).(a) Write a model for the average cost per ton of organic material recycled.(b) Find the average costs of recycling 100 tons and1000 tons of organic material.(c)Determine the limit of the average cost function asapproaches infinity. Explain the meaning of thelimit in the context of the problem.x60x1650,xxxCC1000.x100xCC13.50x45,750,an3n n1n24n4n n122an10n3n n13n16an4nn4nn n12an1nn1nn n12nan1n1n2an1nnan3n1 !3n1 !ann1 !n!an4n212nann23n2an4n1n3ann2n1annn21ann1n21nf x4 4x16x2xf x3 2x4x2xf x3x9x21f xxx22xf x21xf x13x2f x2x1x21f x2x1x2n100101102103anx100101102103104105106f x59.MODELING DATAThe table shows the numbers (in thousands) ofUnited States military reserve personnel in the years2002 through 2008.(Source:U.S. Dept. of Defense)A model for the data is given bywhere represents the year, with correspondingto 2002.(a)Use a graphing utility to create a scatter plot of thedata and graph the model in the same viewing window. How do they compare?(b)Use the model to predict the numbers of reservesin 2009 and 2010.(c)Find the limit of the model as and interpretits meaning in the context of the situation.(d)Is this a good model for predicting future numbersof reserves? Explain.t→t2t2t8R t61.018t21260.640.0578t21,RYearReserves,R(in thousands)20021222200311892004116720051136200611202007111020081100
788Chapter 11Limits and an Introduction to Calculus60.(p. 780)The table showsthe annual payrolls (in millions of dollars)of the legislative branch of the United Statesgovernment for the years 2001 through2008.(Source: U.S. Office of PersonnelManagement)A model for the data is given bywhere represents the year, with corresponding to2001.(a)Use a graphing utility to create a scatter plot of thedata and graph the model in the same viewing window. How do they compare?(b)Use the model to predict the payrolls in 2009 and2010.(c)Find the limit of the model as and interpretits meaning in the context of the situation.(d) Is this a good model for predicting the annual payrolls in future years? Explain.ConclusionsTrue or False?In Exercises 61–64, determine whetherthe statement is true or false. Justify your answer.61.Every rational function has a horizontal asymptote.62.If a rational function has a horizontal asymptote to theright, then the limit of as approaches exists.63.If a sequence converges, then it has a limit.64.When the degrees of the numerator and denominator of a rational function are equal, the limit as goes toinfinity does not exist.65.Think About ItFind functions and such that bothand increase without bound as approachesbut
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