View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.1_Sequences/10.1.43.pg

    1 ## DBsubject('Calculus')
    2 ## DBchapter('Limits and Derivatives')
    3 ## DBsection('Definition of the Derivative')
    4 ## KEYWORDS('calculus', 'derivatives', 'slope')
    5 ## TitleText1('Calculus: Early Transcendentals')
    6 ## EditionText1('2')
    7 ## AuthorText1('Rogawski')
    8 ## Section1('10.1')
    9 ## Problem1('43')
   10 ## Author('Keith Thompson')
   11 ## Institution('W.H.Freeman')
   12 
   13 DOCUMENT();
   14 loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl");
   15 loadMacros("Parser.pl");
   16 loadMacros("freemanMacros.pl");
   17 loadMacros("PGauxiliaryFunctions.pl");
   18 loadMacros("PGgraphmacros.pl");
   19 
   20 #$showPartialCorrectAnswers=1;
   21 
   22 $num=random(2,9);
   23 $den=random(2,13);
   24 $ans=$num/$den;
   25 Context()->texStrings;
   26 BEGIN_TEXT
   27 \{ beginproblem() \}
   28 \{ textbook_ref_exact("Rogawski ET 2e", "10.1","43") \}
   29 $PAR
   30 Determine the limit of the sequence or show that the sequence diverges by using the appropriate Limit Laws or theorems.  If the sequence diverges, enter DIV as your answer.
   31 \[a_n=\frac{$num n^2+n+2}{$den n^2-3}\]
   32 
   33 $PAR
   34 \(\lim\limits_{n\to\infty}a_n = \)  \{ans_rule()\}
   35 END_TEXT
   36 
   37 Context()->normalStrings;
   38 
   39 #ANS(Real($ans)->cmp);
   40 ANS(std_num_str_cmp($ans,['DIV']));
   41 
   42 Context()->texStrings;
   43 SOLUTION(EV3(<<'END_SOLUTION'));
   44 $PAR
   45 $SOL
   46 We have \(a_n=f(n)\), where \(f(x)=\frac{$num x^2+x+2}{$den x^2-3}\). Thus,
   47 
   48 \[\lim_{n\rightarrow \infty} \frac{$num n^2+n+2}{$den n^2-3} = \lim_{x\rightarrow \infty} \frac{$num x^2+x+2}{$den x^2-3} =\frac{$num}{$den}.\]
   49 END_SOLUTION
   50 
   51 ENDDOCUMENT();