qu.1.topic=IF's@

qu.1.1.mode=Multiple Selection@
qu.1.1.name=IF's@
qu.1.1.comment=@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$z1=rint(9);
$z2=rint(9);
$z3=rint(7);
$z4=rint(7);
condition:not(eq($z1,$z2));
condition:not(eq($z3,$z4));
$a=switch($z1, "infinity - infinity",
"0/0","(+infinity)/(-infinity)","(-infinity)/(+infinity)",
"(-infinity)/(-infinity)",
"infinity/infinity","0^0","1^infinity","infinity^0");
$b=switch($z2, "infinity - infinity",
"0/0","(+infinity)/(-infinity)","(-infinity)/(+infinity)",
"(-infinity)/(-infinity)",
"infinity/infinity","0^0","1^infinity","infinity^0");
$c=switch($z3, "infinity + infinity",
"0/infinity","infinity/0","-infinity - infinity",
"0^infinity","1^0","0^1");
$d=switch($z4, "infinity + infinity",
"0/infinity","infinity/0","-infinity - infinity",
"0^infinity","1^0","0^1");@
qu.1.1.uid=17c91dc0-5eb0-469a-b0db-d090602a0fc3@
qu.1.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introductio to Calculus II;
  Topic=L'Hopital's Rule;
  Sub-Topic=Indeterminant forms;
@
qu.1.1.question=<p>Which of the following (there may be more than one!) are indeterminate forms?</p>@
qu.1.1.answer=1, 2@
qu.1.1.choice.1=$a@
qu.1.1.choice.2=$b@
qu.1.1.choice.3=$c@
qu.1.1.choice.4=$d@
qu.1.1.fixed=@

qu.1.2.mode=Multiple Choice@
qu.1.2.name=Lhopy@
qu.1.2.comment=@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=@
qu.1.2.uid=a5599602-bff6-482d-8cc5-b0851aec0256@
qu.1.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=L'Hopital's Rule;
  Sub-Topic=Theory;
@
qu.1.2.question=<p>Which of the following conditions form the <strong>HYPOTHESIS</strong> for the basic form of L'Hopital's Rule?</p>
<p>Let <em>f</em> and <em>g</em> be functions defined on an open interval&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi></mrow></mstyle></math> containing&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>a</mi></mrow></mstyle></math>such that</p>
<p>(i) <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>a</mi></mrow></munder><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>f</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow></mrow><mrow></mrow><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>a</mi></mrow></munder></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mo mathvariant='bold' fontweight='bold' lspace='0.0em' rspace='0.0em'>or</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&pm;</mo></mrow><mrow><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><mi></mi></mrow></mstyle></math></p>
<p>(ii) <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>f</mi><mi>&prime;</mi></msup></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>g</mi><mi>&prime;</mi></msup></mrow></mstyle></math>exist on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi></mrow></mstyle></math></p>
<p>(iii) <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>g</mi><mi>&prime;</mi></msup></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&ne;</mo><mn>0</mn></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mi>I</mi></mrow></mstyle></math></p>
<p>(iv) <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&ne;</mo><mn>0</mn></mrow></mstyle></math>for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mi>I</mi></mrow></mstyle></math></p>
<p>(v) <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>a</mi></mrow></munder><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mrow><msup><mi>f</mi><mi>&prime;</mi></msup></mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mrow><msup><mi>g</mi><mi>&prime;</mi></msup></mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mi></mi></mfrac></mrow></mstyle></math>exists</p>
<p>(vi) <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>a</mi></mrow></munder><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mi>f</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mi></mi></mfrac></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>a</mi></mrow></munder><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mrow><msup><mi>f</mi><mi>&prime;</mi></msup></mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mrow><msup><mi>g</mi><mi>&prime;</mi></msup></mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mi></mi></mfrac></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math></p>@
qu.1.2.answer=2@
qu.1.2.choice.1=all except (vi)@
qu.1.2.choice.2=(i), (ii), (iii), (v)@
qu.1.2.choice.3=(i), (ii), (iv), (v)@
qu.1.2.choice.4=(i), (ii), (v)@
qu.1.2.choice.5=(i), (ii), (v), (vi)@
qu.1.2.choice.6=All of them!@
qu.1.2.fixed=@

qu.2.topic="0/0"@

qu.2.1.question=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><munder accentunder='false'><mrow><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='thinmathspace' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='true' accent='unset'>l</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >im</mo></mrow><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'   fence='unset' separator='unset' lspace='thickmathspace' rspace='thickmathspace' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&rightarrow;</mo><mi    mathvariant='normal' >$a</mi></mrow></munder></mrow></mrow></math>$F</p>@
qu.2.1.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.2.1.allow2d=1@
qu.2.1.maple_answer=limit($f,x=$a);@
qu.2.1.type=formula@
qu.2.1.mode=Maple@
qu.2.1.name="0/0" x approaches a@
qu.2.1.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mi mathvariant='normal'>$a</mi></mrow></munder></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lsqb;</mo></mrow></mstyle></math>$F<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1111111em' stretchy='true'>&rsqb;</mo></mrow></mstyle></math>&nbsp;&nbsp;&nbsp; "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>0</mn><mrow><mn>0</mn></mrow></mfrac></mrow></mstyle></math>"</p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mi mathvariant='normal'>$a</mi></mrow></munder></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lsqb;</mo></mrow></mstyle></math>$Step1<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1111111em' stretchy='true'>&rsqb;</mo></mrow></mstyle></math>&nbsp;by L'Hopital's Rule</p>
<p>= $Step2</p>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$a=rint(5)+1;
$b=$a^2;
$c=$a^3;
$z=rint(4);
$f=switch($z, "(x^2-$a^2)/(x^3-$a^3)", "(x^2-$a^2)/(x^3+$a*x^2-$a^2*x-$a^3)","(x-$a)/(x^2-$a^2)","(x-$a)/($a^2-x^2)");
$t1=switch($z, "2*x", "2*x","1","1");
$b1=switch($z, "3*x^2", "3*x^2+$a*2*x-$a^2", "2*x", "-2*x");
$t2=switch($z, 2*$a, 2*$a, 1, 1);
$b2=switch($z, 3*$b, 4*$b, 2*$a, -2*$a);
$F=maple("printf(MathML[ExportPresentation]($f))");
$Step1=mathml("($t1)/($b1)");
$Step2=mathml("$t2/$b2");@
qu.2.1.uid=958c76a2-6162-4c04-b0b7-7a4d62e862d4@
qu.2.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=L'Hopital's Rule;
  Sub-Topic=Take limit;
@

qu.2.2.question=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math>Evaluate:<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true' accent='true'>&rarr;</mo><mrow><mi>&infin;</mi></mrow></mrow></mrow></munder></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lsqb;</mo></mrow></mstyle></math>$F<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1111111em' stretchy='true'>&rsqb;</mo></mrow></mstyle></math></p>@
qu.2.2.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.2.2.allow2d=1@
qu.2.2.maple_answer=limit($f,x=infinity);@
qu.2.2.type=formula@
qu.2.2.mode=Maple@
qu.2.2.name=sqrt(inf)-sqrt(inf)@
qu.2.2.comment=<p>This is the indeterminant form <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&quot;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&quot;</mo></mrow></mstyle></math>.Start this question by dividing top and bottom by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math>.</p>@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=$a=range(2,5);
$b=range(1,9);
$n=range(4,10);
$m=switch(rint(4),$n-1,$n-1,$n-1,$n-2);
$f=switch(rint(2),"(x^$n+$a*x^$m+$b)^(1/$n)-x", "(x^$n-$a*x^$m+$b)^(1/$n)-x");
$F=maple("printf(MathML[ExportPresentation]($f))");@
qu.2.2.uid=bf926e3c-164c-46a4-9b4b-e260fc545ef4@
qu.2.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus II;
  Topic=L'Hopital's Rule;
  Sub-Topic=Take limit;
@

qu.2.3.question=<p>Evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true' accent='true'>&rarr;</mo><mrow><mi>&infin;</mi></mrow></mrow></mrow></munder></mrow></mstyle></math>$F</p>@
qu.2.3.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.2.3.allow2d=1@
qu.2.3.maple_answer=limit($f,x=infinity);@
qu.2.3.type=formula@
qu.2.3.mode=Maple@
qu.2.3.name=f^ginf@
qu.2.3.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mrow><mi>&infin;</mi></mrow></mrow></munder></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lsqb;</mo></mrow></mstyle></math>$F <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1111111em' stretchy='true'>&rsqb;</mo></mrow></mstyle></math>&nbsp;"$IF1"</p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mrow><mi>&infin;</mi></mrow></mrow></munder></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lsqb;</mo></mrow></mstyle></math>$Step1<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1111111em' stretchy='true'>&rsqb;</mo></mrow></mstyle></math>&nbsp;"$IF2"</p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mrow><mi>&infin;</mi></mrow></mrow></munder></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lsqb;</mo></mrow></mstyle></math>$Step2<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1111111em' stretchy='true'>&rsqb;</mo></mrow></mstyle></math>&nbsp;by L'Hopital's Rule</p>
<p>= $Step3</p>@
qu.2.3.editing=useHTML@
qu.2.3.solution=@
qu.2.3.algorithm=$a=rint(2,5);
$b=rint(1,9);
$z=rint(4);
$ab=$a*$b;
$f=switch($z, "(1+$a*x)^(1/x)","(1+$a/x)^($b*x)","(1-$a/x)^($b*x)","(e^x+$a)^(1/x)");
$step1=switch($z,"e^(ln(1+$a*x)/x)","e^($b*ln(1+$a/x)/(1/x))","e^($b*ln(1-$a/x)/(1/x))","e^(ln(e^x+$a)/x)");
$step2=switch($z,"e^($a/(1+$a*x))", "e^($a*$b/(1+$a/x))","e^(-$a*$b/(1-$a/x))","e^((e^x)/(e^x+$a))");
$step3=switch($z, "1", "e^($ab)", "e^(-$ab)", "e^1");
$if1=switch($z, "infinity^infinity", "1^infinity", "1^infinity", "infinity^0");
$if2=switch($z, "infinity/infinity", "O/O", "O/O", "infinity/infinity");
$F=mathml("$f","no simplify");
$Step1=mathml("$step1");
$Step2=mathml("$step2");
$Step3=mathml("$step3");
$IF1=mathml("$if1");
$IF2=mathml("$if2");@
qu.2.3.uid=fcb38562-c744-4846-8fd0-a4080e2d5f53@
qu.2.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus II;
  Topic=L'Hopital's Rules;
  Sub-Topic=Take limit;
@

qu.2.4.question=<p>Evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mn>0</mn></mrow></munder></mrow></mstyle></math>$F</p>@
qu.2.4.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.2.4.allow2d=1@
qu.2.4.maple_answer=limit($f,x=0,right);@
qu.2.4.type=formula@
qu.2.4.mode=Maple@
qu.2.4.name=f^gx approaches 0@
qu.2.4.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mn>0</mn></mrow></munder></mrow></mstyle></math>&nbsp;$F&nbsp; "$IF1"</p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mn>0</mn></mrow></munder></mrow></mstyle></math>$Step1&nbsp; "$IF2"</p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mn>0</mn></mrow></munder></mrow></mstyle></math>$Step2 by L'Hopital's Rules</p>
<p>= $Step3</p>@
qu.2.4.editing=useHTML@
qu.2.4.solution=@
qu.2.4.algorithm=$a=rint(2,6);
$b=rint(1,10);
$z=rint(5);
$f=switch($z, "(1+$a*x)^($b/x)","(1-$a*x)^($b/x)","($a*x)^x","x^($b*x)","(1-exp(x))^(x)");
$step1=switch($z, "e^(($b*ln(1+$a*x))/$x)", "e^(($b*ln(1-$a*x))/$x)", "e^(ln($a*x)/(1/x))", "e^($b*ln(x)/(1/x))", "e^(ln(1-e^x)/(1/x))");
$step2=switch($z, "e^(($b*$a)/(1+$a*x))", "e^((-$b*$a)/(1-$a*x))", "e^((1/x)/(-1/x^2))","e^(($b/x)/(-1/x^2))", "e^((-e^x)/(1-e^x)/(-1/x^2))");
$step3=switch($z, "e^(($b*$a))", "e^(-$b*$a)", "1", "1", "1");
$ANSWER=maple("limit($f,x=0,right)");
$F=maple("printf(MathML[ExportPresentation]($f))");
$Step1=mathml("$step1");
$Step2=mathml("$step2");
$Step3=mathml("$step3");
$if2=switch($z, "O/O", "O/O", "infinity/infinity","infinity/infinity","infinity/infinity");
$if1=switch($z, "1^infinity", "1^infinity", "0^0","0^0","0^0");
$IF1=mathml("$if1");
$IF2=mathml("$if2");@
qu.2.4.uid=e852914c-c4fa-4313-92ef-0f028533cefc@
qu.2.4.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus II;
  Topic=L'Hopital's Rule;
  Sub-Topic=Take limit;
@

qu.2.5.question=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><munder accentunder='false'><mrow><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='thinmathspace' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='true' accent='unset'>l</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >im</mo></mrow><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'   fence='unset' separator='unset' lspace='thickmathspace' rspace='thickmathspace' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&rightarrow;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi></mrow></munder></mrow></mrow></math>$F</p>@
qu.2.5.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.2.5.allow2d=1@
qu.2.5.maple_answer=limit($f,x=-$a);@
qu.2.5.type=formula@
qu.2.5.mode=Maple@
qu.2.5.name="0/0" x approaches -a@
qu.2.5.comment=<p>This "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>0</mn><mrow><mn>0</mn></mrow></mfrac></mrow></mstyle></math>" limit is set up perfectly for L'Hopital's Rule.</p>@
qu.2.5.editing=useHTML@
qu.2.5.solution=@
qu.2.5.algorithm=$a=rint(5)+1;
$d=rint(4)+6;
$b=$a^2;
$c=$a^3;
$z=rint(5);
$f= switch($z, "(x^2-$a^2)/(x^3+$a^3)","(x+$a)/(x^2-$a^2)","(x+$a)/($a^2-x^2)","(x+$a)/((x^2+($a+$d)*x+$a*$d))","(x^2+($a+$d)*x+$a*$d)/(x^2-$a^2)");
$F=maple("printf(MathML[ExportPresentation]($f))");@
qu.2.5.uid=06a726c5-d720-47a3-a9a5-4b4bd266fd91@
qu.2.5.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=L'Hopital's Rule;
  Sub-Topic=Take limit;
@

qu.2.6.question=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><munder accentunder='false'><mrow><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='thinmathspace' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='true' accent='unset'>l</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >im</mo></mrow><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'   fence='unset' separator='unset' lspace='thickmathspace' rspace='thickmathspace' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&rightarrow;</mo><mi    mathvariant='normal' >$a</mi></mrow></munder></mrow></mrow></math>$F</p>@
qu.2.6.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.2.6.allow2d=1@
qu.2.6.maple_answer=limit($f,x=$a);@
qu.2.6.type=formula@
qu.2.6.mode=Maple@
qu.2.6.name="0/0withSIN"@
qu.2.6.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mi mathvariant='normal'>$a</mi></mrow></munder></mrow></mstyle></math>$F "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>0</mn><mrow><mn>0</mn></mrow></mfrac></mrow></mstyle></math>"</p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mi mathvariant='normal'>$a</mi></mrow></munder></mrow></mstyle></math>$Step1 by L'Hopital's Rule</p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn></mrow></mstyle></math></p>@
qu.2.6.editing=useHTML@
qu.2.6.solution=@
qu.2.6.algorithm=$a=rint(9)+2;
$z=rint(2);
$f=switch($z,"sin(x-$a)/(x-$a)","(x-$a)/sin(x-$a)");
$step1=switch($z,"cos(x-$a)/1","1/cos(x-$a)");
$F=mathml($f);
$Step1=mathml("$step1");@
qu.2.6.uid=17072dbb-d94e-4c16-a7f3-4471ae896c24@
qu.2.6.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=L'Hopital's Rule;
  Sub-Topic=Take limit;
@

qu.2.7.question=<p>Evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mn>0</mn></mrow></munder></mrow></mstyle></math>$F.</p>@
qu.2.7.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.2.7.allow2d=1@
qu.2.7.maple_answer=limit($f,x=0);@
qu.2.7.type=formula@
qu.2.7.mode=Maple@
qu.2.7.name=(exp(x)-1)/x@
qu.2.7.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mn>0</mn></mrow></munder></mrow></mstyle></math>$F&nbsp;&nbsp;&nbsp;&nbsp; "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>0</mn><mrow><mn>0</mn></mrow></mfrac></mrow></mstyle></math>"</p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mn>0</mn></mrow></munder></mrow></mstyle></math>$Step1&nbsp;&nbsp; $if&nbsp;&nbsp;&nbsp;&nbsp; by L'Hopital's Rule</p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mn>0</mn></mrow></munder></mrow></mstyle></math>&nbsp;$Step2&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $reason</p>
<p>= $Step3</p>@
qu.2.7.editing=useHTML@
qu.2.7.solution=@
qu.2.7.algorithm=$z=rint(2);
$f=switch($z, "(e^(x)-1)/x","(e^x-x-1)/(x^2)");
$step1=switch($z, "e^x/1" , "(e^x-1)/(2*x)");
$step2=switch($z, "e^x", "e^x/2");
$step3=switch($z, "1", "1/2");
$reason=switch($z, "", "by L'Hopital's Rule");
$if=switch($z, "" , '"0/0"');
$F=maple("printf(MathML[ExportPresentation]($f))");
$Step1=mathml("$step1");
$Step2=mathml("$step2");
$Step3=mathml("$step3");@
qu.2.7.uid=6153fcc8-a38d-4a14-ba83-5dc8f2d46e24@
qu.2.7.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=L'Hopital's Rule;
  Sub-Topic=Take limit;
@

qu.2.8.question=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math>Evaluate:<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&infin;</mi></mrow></mrow></munder></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lsqb;</mo></mrow></mstyle></math>$F<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1111111em' stretchy='true'>&rsqb;</mo></mrow></mstyle></math></p>
<p><strong>Hint: </strong>When <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msqrt><mrow><mi>x</mi></mrow></msqrt><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math></p>@
qu.2.8.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.2.8.allow2d=1@
qu.2.8.maple_answer=limit($f,x=-infinity);@
qu.2.8.type=formula@
qu.2.8.mode=Maple@
qu.2.8.name=-sqrt(inf)+sqrt(inf)@
qu.2.8.comment=<p>This is the indeterminant form "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow><mrow><mi></mi></mrow></mstyle></math>". To start this question divide top and bottom by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math>.</p>@
qu.2.8.editing=useHTML@
qu.2.8.solution=@
qu.2.8.algorithm=$a=range(2,5);
$b=range(1,9);
$n=range(4,10,2);
$m=switch(rint(4), $n-1,$n-1,$n-1,$n-2);
$f=switch(rint(2),"(x^$n+$a*x^$m+$b)^(1/$n)+x",
"(x^$n-$a*x^$m+$b)^(1/$n)+x");

$F=maple("printf(MathML[ExportPresentation]($f))");@
qu.2.8.uid=5336e4ff-8bca-484b-8b45-4e4158d5b0fe@
qu.2.8.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus II;
  Topic=L'Hopital's Rule;
  Sub-Topic=Take limit;
@

qu.2.9.question=<p>Evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mn>0</mn></mrow></munder></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow></mstyle></math>$F<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math>.</p>@
qu.2.9.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.2.9.allow2d=1@
qu.2.9.maple_answer=limit($f,x=0);@
qu.2.9.type=formula@
qu.2.9.mode=Maple@
qu.2.9.name=inf-inf fractions@
qu.2.9.comment=<p>Make a common denominator to combine the fractions, then use L'Hopital's Rule</p>@
qu.2.9.editing=useHTML@
qu.2.9.solution=@
qu.2.9.algorithm=$z=rint(4);
$f=switch($z, "1/(exp(x)-1)-1/x" ,"1/x-1/(exp(x)-1)","1/sin(x)-1/x","1/ln(x+1)-1/x");
$F=maple("printf(MathML[ExportPresentation]($f))");@
qu.2.9.uid=c8bac440-3f58-4229-80b7-86fe850c9adf@
qu.2.9.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=L'Hopital's Rule;
  Sub-Topic=Take limit;
@

qu.2.10.question=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><munder accentunder='false'><mrow><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='thinmathspace' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='true' accent='unset'>l</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >im</mo></mrow><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'   fence='unset' separator='unset' lspace='thickmathspace' rspace='thickmathspace' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&rightarrow;</mo><mn    mathvariant='normal' >0</mn></mrow></munder></mrow></mrow></math>$F</p>@
qu.2.10.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.2.10.allow2d=1@
qu.2.10.maple_answer=limit($f,x=0);@
qu.2.10.type=formula@
qu.2.10.mode=Maple@
qu.2.10.name=Trig/Trig or Trig/Poly@
qu.2.10.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mn>0</mn></mrow></munder></mrow></mstyle></math>$F "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>0</mn><mrow><mn>0</mn></mrow></mfrac></mrow></mstyle></math>"</p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mn>0</mn></mrow></munder></mrow></mstyle></math>$Step1&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$if&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;by L'Hopital's Rule</p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mn>0</mn></mrow></munder></mrow></mstyle></math>$Step2 $reason</p>
<p>= $Step3</p>@
qu.2.10.editing=useHTML@
qu.2.10.solution=@
qu.2.10.algorithm=$a=rint(5)+2;
$b=rint(5)+2;
$bs=$b*$b;
$z=rint(6);
$f=switch($z , "sin($b*x)/tan($a*x)","(1-cos($b*x))/x","(1-cos($b*x))/x^2","x/sin($a*x)","tan(x)/tan($b*x)","sin($b*x)/x");
$step1=switch($z , "($b*cos($b*x))/($a*(sec($a*x))^2)","($b*sin($b*x))/1","($b*sin($b*x))/2*x","1/($a*cos($a*x))","(sec(x))^2/($b*(sec($b*x))^2)","$b*cos($b*x)/1");
$step2=switch($z , "($b*cos($b*x)*(cos($a*x))^2)/($a)","($b*sin($b*x))","($bs*cos($b*x))/2","1/($a*cos($a*x))","(cos($b*x))^2/($b*(cos(x))^2)","$b*cos($b*x)");
$step3=switch($z , "($b)/($a)","0","($bs)/2","1/($a)","1/($b)","$b");
$reason=switch($z, "", "", "by L'Hopitals Rule", "", "", "");
$if=switch($z, "", "", '"0/0"', "", "", "");
$F=mathml($f);
$Step1=mathml("$step1");
$Step2=mathml("$step2");
$Step3=mathml("$step3");@
qu.2.10.uid=bcd2e6e9-9f42-4283-afa6-abbc26b3601e@
qu.2.10.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=L'Hopital's Rule;
  Sub-Topic=Take limit;
@

qu.3.topic=not an IF@

qu.3.1.question=<p>Evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true' accent='true'>&rarr;</mo><mrow><mi>&infin;</mi></mrow></mrow></mrow></munder></mrow></mstyle></math>$F.</p>@
qu.3.1.maple=evalb(($ANSWER)=($RESPONSE));@
qu.3.1.allow2d=1@
qu.3.1.maple_answer=limit($f,x=infinity);@
qu.3.1.type=formula@
qu.3.1.mode=Maple@
qu.3.1.name=not an IF@
qu.3.1.comment=<p>If your answer is correct, GOOD!If you applied L'Hopital's Rule here, you probably got the wrong answer. This question does NOT involve an indeterminate form. BE CAREFUL!</p>@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$f=switch(rint(7), "x*exp(x)","(1/x)/exp(x)","sin(1/x)/x","x^x","(1/x)^x",
"(1+x)^x","(1+1/x)^(1/x)");
$F=maple("printf(MathML[ExportPresentation]($f))");@
qu.3.1.uid=153a64dd-6d37-41d3-8933-263abecc2950@
qu.3.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=L'Hopital's Rule;
  Sub-Topic=Not an indeterminant form;
@

