Hello,

I authored the following problem, but WeBWorK always gives the "cannot generate enough valid points for comparison" on the last formula (n!). I've tried giving it specific integer points to test but then it tells me it can't test those points. What do I do?

DOCUMENT(); loadMacros( "PGstandard.pl", # Standard macros for PG language "MathObjects.pl", "contextFraction.pl", "PGML.pl", "scaffold.pl", "PGchoicemacros.pl", "PGstandard.pl", "PGunion.pl", "PGnumericalmacros.pl", "PGstatisticsmacros.pl", "MathObjects.pl", "parserPopUp.pl", "PGML.pl", "unionTables.pl", "niceTables.pl", "PGcourse.pl", "PGchoicemacros.pl", "answerHints.pl", "weightedGrader.pl", "parserRadioButtons.pl", "parserNumberWithUnits.pl", "randomizers.pl" ); ########################### # Setup Context("Numeric"); Context()->variables->add(n => 'Real'); Context()->{format}{number} = "%.9f#"; $a = random(2,5); $b = random(2,5); $fact = Formula("n!"); $log = Formula("ln(n)"); $pow = Formula("n^($a)"); $powlog = Formula("n^($a)*ln(n)"); $exp = Formula("$b^n"); $powexp = Formula("n^($a)*$b^n"); $ans1 = $log; $ans2 = $pow; $ans3 = $powlog; $ans4 = $exp; $ans5 = $powexp; $ans6 = $fact; $seed = random(1,5); if ($seed == 1) { $func1 = $fact; $func2 = $log; $func3 = $pow; $func4 = $powlog; $func5 = $exp; $func6 = $powexp; } elsif ($seed == 2) { $func1 = $powexp; $func2 = $fact; $func3 = $powlog; $func4 = $exp; $func5 = $log; $func6 = $pow; } elsif ($seed == 3) { $func1 = $log; $func2 = $exp; $func3 = $powexp; $func4 = $powlog; $func5 = $fact; $func6 = $pow; } elsif ($seed == 4) { $func1 = $pow; $func2 = $log; $func3 = $fact; $func4 = $powlog; $func5 = $exp; $func6 = $powexp; } elsif ($seed == 5) { $func1 = $powlog; $func2 = $fact; $func3 = $pow; $func4 = $exp; $func5 = $powexp; $func6 = $log; } ########################### # Main text BEGIN_PGML A big part of having good intuition for the Ratio Test is knowing which functions grow faster than which other functions. Recall that we say [`a_n`] is *dominated* by [`b_n`] (written [`a_n\prec b_n`]) if [``\lim\limits_{n\to\infty} \dfrac{a_n}{b_n}=0``] (or, equivalently, if [``\lim\limits_{n\to\infty} \dfrac{b_n}{a_n}=\infty``]). Essentially it means that [`a_n`] grows more slowly than [`b_n`]. Rank the following six functions in order from *slowest-growing* to *fastest-growing*: [```[$func1],\quad [$func2],\quad [$func3],\quad [$func4],\quad [$func5],\quad [$func6]```] [_]{$ans1} (slowest) [`\prec`] [_]{$ans2} [`\prec`] [_]{$ans3} [`\prec`] [_]{$ans4} [`\prec`] [_]{$ans5} [`\prec`] [_]{$ans6} (fastest) END_PGML ############################ # Answer evaluation ############################ # Solution BEGIN_PGML_SOLUTION The slowest-growing is [`[$ans1]`]. Next comes [`[$ans2]`], since powers grow faster than logarithms. Next comes [`[$ans3]`], since multiplying by an extra logarithm makes the function grow faster. Next comes [`[$ans4]`], since exponentials grow faster than powers. Next comes [`[$ans5]`], since multiplying by an extra power makes the function grow faster. Finally, the fastest-growing is [`[$ans6]`]. Hence we have: [```[$ans1]\prec[$ans2]\prec[$ans3]\prec[$ans4]\prec[$ans5]\prec [$ans6]```] END_PGML_SOLUTION COMMENT('MathObject version. Uses PGML.'); ENDDOCUMENT();