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| 1 : | aubreyja | 2584 | ## 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(); |
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