#Problem APEX 6.7.31

DOCUMENT();

# Load whatever macros you need for the problem
loadMacros(
  "PGstandard.pl",
  "MathObjects.pl",
  "PGcourse.pl"
);

## DBsubject(Calculus - single variable)
## DBchapter(Applications of differentiation)
## DBsection(Indeterminate forms and L'Hopital's rule)
## Institution(Valdosta State University)
## Author(S. V. Ault)
## Level(3)
## MO(1)
## TitleText1('APEX Calculus')
## AuthorText1('Hartman')
## EditionText1('3.0')
## Section1('6.7')
## Problem1('31')

$showPartialCorrectAnswers = 1;

$a = random(2,15,1);

$c = "0^{+}";

$num = "$a e^{x^{-1}}";
$den = "x^{-1}";

$f = "$a x e^{1/x}";

$dnum = "$a e^{x^{-1}} (-x^{-2})";
$dden = "-x^{-2}";

$ans = "INF";


TEXT(beginproblem());

BEGIN_TEXT
Evaluate the limit, using L'H\{ MODES(HTML=>"&ocirc;",TeX=>"\^o") \}pital's Rule. 
$BR
Enter $BBOLD INF $EBOLD for \(\infty\), $BBOLD -INF $EBOLD 
for \(-\infty\), or $BBOLD DNE $EBOLD if the limit does not exist,
but is neither \(\infty\) nor \(-\infty\).
$PAR
\( \displaystyle 
  \lim_{x\to $c} $f =
\) \{ans_rule(10) \}
END_TEXT

ANS(Compute($ans)->cmp());

Context()->normalStrings;
BEGIN_SOLUTION
First re-express the function as
a fraction.
\[ 
   \lim_{x \to $c} $f =
  \lim_{x \to $c} \frac{$num}{$den}
  = \lim_{x \to $c} \frac{$dnum}{$dden}
\]
\[
  = \lim_{x \to $c} $a e^{x^{-1}}
  = $a e^{\infty} = \infty.
\]
END_SOLUTION

ENDDOCUMENT();
