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    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('51')
   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 $ans=ln($num/$den);
   25 Context()->texStrings;
   26 BEGIN_TEXT
   27 \{ beginproblem() \}
   28 \{ textbook_ref_exact("Rogawski ET 2e", "10.1","51") \}
   29 $PAR
   30 
   31 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.
   32 \[c_n=\ln\left(\frac{$num n-7}{$den n+4}\right)\]
   33 
   34 $PAR
   35 \(\lim\limits_{n\to\infty}c_n = \)  \{ans_rule()\}
   36 END_TEXT
   37 
   38 Context()->normalStrings;
   39 
   40 #ANS(Real($ans)->cmp);
   41 ANS(std_num_str_cmp($ans,['DIV']));
   42 Context()->texStrings;
   43 SOLUTION(EV3(<<'END_SOLUTION'));
   44 $PAR
   45 $SOL
   46 Because \(f(x)=\ln x\) is a continuous function, it follows that
   47 
   48 
   49 \[\lim_{n\rightarrow \infty} c_n = \lim_{x\rightarrow \infty}\ln\left(\frac{$num x-7}{$den x+4}\right)=\ln\left(\lim_{x\rightarrow \infty}\frac{$num x-7}{$den x+4}\right)=\ln \left(\frac{$num}{$den}\right).\]
   50 END_SOLUTION
   51 
   52 ENDDOCUMENT();