qu.1.topic=NoProblemLimits@

qu.1.1.question=<p>Find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi mathvariant='normal'>$b</mi></mrow></munder></mrow></math>$F.</p>@
qu.1.1.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.1.1.allow2d=1@
qu.1.1.maple_answer=limit($f,x=$b)@
qu.1.1.type=formula@
qu.1.1.mode=Maple@
qu.1.1.name=NoProblemFunctions@
qu.1.1.comment=@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$a=rint(-5,5);
$b=rint(-5,-2);
$n=rint(1,3);
condition: ne($a,0);
$f=switch(rint(6),"exp(($a)*x)","sin(Pi*($a)*x)","cos(Pi*($a)*x)","x^($n)", "1/sqrt(x^2+($a)^2)","(x^2+($a)^2)^(1/3)","x/exp(($a)*x)","x/(x^2+($a)^2)");
$F=maple("printf(MathML[ExportPresentation]($f))");@
qu.1.1.uid=c3341094-7b8a-4be6-bf4f-6d552541a0a4@
qu.1.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculuc I;
  Topic=Intuitive Limts;
  Sub-Topic=Basic Limits;
  Difficulty=Easy;
@

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

qu.2.1.question=<p>Evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi mathvariant='normal'>$c</mi></mrow></munder></mrow></math>$F.</p>@
qu.2.1.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.2.1.allow2d=1@
qu.2.1.maple_answer=if $z = 0 then
simplify(limit((surd(x,3)+($a))/(x+($a)^3),x=-($a)^3))
else
simplify(limit((x-($a)^3)/(surd(x,3)-($a)),x=($a)^3))
end if@
qu.2.1.type=formula@
qu.2.1.mode=Maple@
qu.2.1.name=(x^(1/3) +a)/(x+a^3) etc@
qu.2.1.comment=<p>This is a "<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mn>0</mn><mrow><mn>0</mn></mrow></mfrac></mrow></math>" limit. Here you should factor, remembering to use the sum/difference of cubes formula.</p>@
qu.2.1.editing=useHTML@
qu.2.1.hint.1=Factor.@
qu.2.1.hint.2=Remember <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi>a</mi><mrow><mn>3</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>b</mi><mrow><mn>3</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>b</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><msup><mi>a</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>ab</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>b</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi>a</mi><mrow><mn>3</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>b</mi><mrow><mn>3</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>b</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><msup><mi>a</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>ab</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>b</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></math>.@
qu.2.1.solution=@
qu.2.1.algorithm=$a=rint(1,5);
$z = rint(2);
$f=switch($z, "(x^(1/3)+($a))/(x+($a)^3)", "(x-($a)^3)/(x^(1/3)-($a))");
$F=maple("printf(MathML[ExportPresentation]($f))");
$c=switch($z, -($a)^3, ($a)^3);@
qu.2.1.uid=7b0cea6c-974b-41b4-8688-967bf5a361f4@
qu.2.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Intuitive Limits;
  Sub-Topic="0/0";
  Difficulty=Medium;
@

qu.2.2.question=<p>Evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi mathvariant='normal'>$a</mi></mrow></munder></mrow></math>$F.</p>@
qu.2.2.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.2.2.allow2d=1@
qu.2.2.maple_answer=limit($f,x=$a)@
qu.2.2.type=formula@
qu.2.2.mode=Maple@
qu.2.2.name=(x^2-a^2)/(x-a),(x+a)/(x^3+a^3)etc@
qu.2.2.comment=<p>This is a "<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mn>0</mn><mrow><mn>0</mn></mrow></mfrac></mrow></math>" limit, here you should factor. Remember your difference of squares and your sum/difference of cubes formulas.</p>@
qu.2.2.editing=useHTML@
qu.2.2.hint.1=Factor.@
qu.2.2.hint.2=Remember:&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi>a</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>b</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>b</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>b</mi></mrow></mfenced></mrow></math>&nbsp;, &nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi>a</mi><mrow><mn>3</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>b</mi><mrow><mn>3</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi>a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>b</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><msup><mi>a</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>ab</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>b</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi>a</mi><mrow><mn>3</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>b</mi><mrow><mn>3</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>b</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><msup><mi>a</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>ab</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>b</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></math>.@
qu.2.2.solution=@
qu.2.2.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
$z = rint(4);
condition: ne($a,0);
condition:ne($a,$b);
$f = switch($z, "expand(x^2-($a)^2)/(x-($a))",
"(($a)-x)/(x^2-($a)^2)", "(x^3-($a)^3)/(x-($a))", "expand((x-($a))*(x+($b)))/expand((x-($a))*(x-($b)))");
$F=maple("printf(MathML[ExportPresentation]($f))");@
qu.2.2.uid=37d42098-5e95-4194-8190-45a8d5bb86ae@
qu.2.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Intuitive Limtis;
  Sub-Topic="0/0";
  Difficulty=Medium;
@

qu.2.3.question=<p>Evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi mathvariant='normal'>$as</mi></mrow></munder></mrow></math>$F.</p>@
qu.2.3.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.2.3.allow2d=1@
qu.2.3.maple_answer=if $f=(x-($a)^2)/(x^(1/2)-($a)) then simplify(limit((x-($a)^2)/(sqrt(x)-($a)),x=($a)^2)) else simplify(limit((sqrt(x)-($a))/(x-($a)^2),x=($a)^2)) end if@
qu.2.3.type=formula@
qu.2.3.mode=Maple@
qu.2.3.name=(x^(1/2) +a)/(x+a^2) etc@
qu.2.3.comment=<p>This is a "<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mn>0</mn><mrow><mn>0</mn></mrow></mfrac></mrow></math>" limit. Use difference of squares to factor <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$as</mi></mrow></math>and then cancel.</p>@
qu.2.3.editing=useHTML@
qu.2.3.hint.1=Factor.@
qu.2.3.solution=@
qu.2.3.algorithm=$a=rint(1,5);
$as=($a)^2;
$f=switch(rint(2),"(x^(1/2)-($a))/(x-($a)^2)",
"(x-($a)^2)/(x^(1/2)-($a))");
$F=maple("printf(MathML[ExportPresentation]($f))");@
qu.2.3.uid=3e78db9d-445e-419f-b3cf-1cad92f7e348@
qu.2.3.info=  Author=Jack Weiner, Gord Clement;
  Topic=Intuitive Limits;
  Course=Introduction to Calculus I;
  Sub-Topic="0/0";
  Difficulty=Medium;
@

qu.2.4.question=<p>Evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi mathvariant='normal'>$a</mi></mrow></munder></mrow></math>$F.</p>@
qu.2.4.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.2.4.allow2d=1@
qu.2.4.maple_answer=limit($f,x=$a)@
qu.2.4.type=formula@
qu.2.4.mode=Maple@
qu.2.4.name=(sqrt(x-a+b^2)-b^2)/(x-a)@
qu.2.4.comment=<p>This is a "<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mn>0</mn><mrow><mn>0</mn></mrow></mfrac></mrow></math>" limit. Here you should multiple the top and bottom by <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msqrt><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$c</mi></mrow></mfenced></mrow></msqrt><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></math>.</p>@
qu.2.4.editing=useHTML@
qu.2.4.hint.1=Rationalize the $place.@
qu.2.4.solution=@
qu.2.4.algorithm=$a=rint(1,8);
$b=rint(1,8);
$z=rint(2);
condition: ne($a,($b)^2);
$f=switch($z,"(sqrt(x-($a)+($b)^2)-($b))/(x-($a))",
"(x-($a))/(sqrt(x-($a)+($b)^2)-($b))");
$F=maple("printf(MathML[ExportPresentation]($f))");
$c = -($a)+($b)^2;
$place=switch($z, "numerator", "denominator");@
qu.2.4.uid=439c4a1c-ca39-4532-b518-f8302a4443ff@
qu.2.4.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Intuitive Limits;
  Sub-Topic="0/0";
  Difficulty=Medium;
@

qu.3.topic=One sided limits@

qu.3.1.question=<p>Find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true' minsize='20px' maxsize='20px' >&lpar;</mo></mrow></math>$F<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true' minsize='20px' maxsize='20px' >&rpar;</mo></mrow></math>.</p>
<p>("floor" is the Maple name for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']' separators=','><mrow><mfenced open='[' close=']' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>greatest integer less than or equal to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>.)</p>@
qu.3.1.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.3.1.allow2d=1@
qu.3.1.maple_answer=limit($f,x=($a),right)@
qu.3.1.type=formula@
qu.3.1.mode=Maple@
qu.3.1.name=FloorRIGHT@
qu.3.1.comment=<p>Tip: Since we&nbsp;are looking at the limit from the right, try plugging in numbers just bigger than <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math>, such as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$tip</mi></mrow></math>.</p>
<p>Here is a plot of the function.</p>
<p>$plot</p>@
qu.3.1.editing=useHTML@
qu.3.1.hint.1=Since we are looking at the limit from the right, try plugging in numbers just bigger than <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math>.@
qu.3.1.solution=@
qu.3.1.algorithm=$a=rint(-2,2);
$b=switch(rint(4),-2,-1,1,2);
$f=switch(rint(3),"floor(($b)*x)","floor(x)-x",
"x-floor(x)");
$F=maple("printf(MathML[ExportPresentation]($f))");

$tip=$a+0.1;
$plot=plotmaple("plot($f,x=-abs($a)-1..abs($a)+1,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.3.1.uid=9fb15b83-2899-4cad-bb94-d1e9e808a6d7@
qu.3.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Intuitive Limtis;
  Sub-Topic=One-sided limits;
  Difficulty=Easy;
@

qu.3.2.question=<p>Find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true' minsize='20px' maxsize='20px'>&lpar;</mo></mrow></math>$F<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true' minsize='20px' maxsize='20px' >&rpar;</mo></mrow></math>.</p>
<p>("floor" is the Maple name for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']' separators=','><mrow><mfenced open='[' close=']' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>greatest integer less than or equal to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>.)</p>@
qu.3.2.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.3.2.allow2d=1@
qu.3.2.maple_answer=limit($f,x=($a),left)@
qu.3.2.type=formula@
qu.3.2.mode=Maple@
qu.3.2.name=FloorLEFT@
qu.3.2.comment=<p>Tip: Since we are looking at the limit from the left, try plugging in numbers just smaller than <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math>, such as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$tip</mi></mrow></math>.</p>
<p>&nbsp;</p>
<p>Here is a plot of the function.</p>
<p>$plot</p>@
qu.3.2.editing=useHTML@
qu.3.2.hint.1=Since we are looking at the limit from the left, try plugging in numbers just smaller than <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math>.@
qu.3.2.solution=@
qu.3.2.algorithm=$a=rint(-2,2);
$b=switch(rint(4),-2,-1,1,2);
$f=switch(rint(3),"floor(($b)*x)","floor(x)-x",
"x-floor(x)");
$F=maple("printf(MathML[ExportPresentation]($f))");

$plot=plotmaple("plot($f,x=-abs($a)-1..abs($a)+1,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");
$tip=$a-0.1;@
qu.3.2.uid=833fa881-47d3-4bff-99da-e681b4cf10f2@
qu.3.2.info=  Author=Jack Weiner, Gord Clement;
  Topic=Intuitive Limits;
  Sub-Topic=One-sided limits;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
@

qu.3.3.question=<p>Let <em>$F.&nbsp;</em></p>
<p>Find<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder></mrow><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math></p>@
qu.3.3.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.3.3.allow2d=1@
qu.3.3.maple_answer=limit($f,x=$a,left)@
qu.3.3.type=formula@
qu.3.3.mode=Maple@
qu.3.3.name=LeftHandLimit@
qu.3.3.comment=<p>$plot</p>@
qu.3.3.editing=useHTML@
qu.3.3.solution=@
qu.3.3.algorithm=$a=rint(-3,5);
$b=rint(-5,5);
$z=rint(9);
$g=switch($z,"sin(x-($a))","cos(x-($a))","tan(x-($a))",
0,1,"(x-($a))+($b)","(x-($a))^2+($b)","(x-($a))^3+($b)","exp(x)+($a)");
$h=switch(rint(9),"sin(x-($a))","cos(x-($a))","tan(x-($a))",
0,1,"(x-($a))+($b)","(x-($a))^2+($b)","(x-($a))^3+($b)","exp(x)+($a)");
condition:ne($g,$h);
$m=maple("
f:=piecewise(x < $a, $g, x >= $a, $h):
convert(f,string), MathML[ExportPresentation](F(x)=f)");
$f=switch(0,$m);
$F=switch(1,$m);
$plot=plotmaple("plot($f,x=-abs($a)-3..abs($a)+3,y=-10..10,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.3.3.uid=316a2c8d-8216-438a-b378-25b4fb7fddfa@
qu.3.3.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Intuitive Limits;
  Sub-Topic=One-sided limits;
  Difficulty=Easy;
@

qu.3.4.question=<p>Let $F.&nbsp;</p>
<p>Find<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder></mrow><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>.</p>@
qu.3.4.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.3.4.allow2d=1@
qu.3.4.maple_answer=limit($f,x=$a,right)@
qu.3.4.type=formula@
qu.3.4.mode=Maple@
qu.3.4.name=RightHandLimit@
qu.3.4.comment=<p>Here is a plot of the function.</p>
<p>$plot</p>@
qu.3.4.editing=useHTML@
qu.3.4.solution=@
qu.3.4.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
$z=rint(9);
$g=switch($z,"sin(x-($a))","cos(x-($a))","tan(x-($a))",
0,1,"(x-($a))+($b)","(x-($a))^2+($b)","(x-($a))^3+($b)","exp(x)+($a)");
$h=switch(rint(9),"sin(x-($a))","cos(x-($a))","tan(x-($a))",
0,1,"(x-($a))+($b)","(x-($a))^2+($b)","(x-($a))^3+($b)","exp(x)+($a)");
condition:ne($g,$h);
$m=maple("
f:=piecewise(x < $a, $g, x >= $a, $h):
convert(f,string),MathML[ExportPresentation](F(x)=(f))");
$f=switch(0, $m);
$F=switch(1,$m);

$plot=plotmaple("plot($f,x=-abs($a)-2..abs($a)+2,y=-10..10,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.3.4.uid=e9b8573b-6484-480c-8f17-16e44ccd092f@
qu.3.4.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Intuitive Limits;
  Sub-Topic=One-sided limits;
  Difficulty=Easy;
@

qu.3.5.question=<p>Find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder></mrow></math>$F.</p>@
qu.3.5.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.3.5.allow2d=1@
qu.3.5.maple_answer=$ANS;@
qu.3.5.type=formula@
qu.3.5.mode=Maple@
qu.3.5.name=|x-a|/(x-a)RIGHT@
qu.3.5.comment=<p>When <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi mathvariant='normal'>$a</mi></mrow></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></math>.</p>
<p>&nbsp;</p>
<p>Here is a plot of the function.</p>
<p>$plot</p>@
qu.3.5.editing=useHTML@
qu.3.5.hint.1=When <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$a</mi></mrow></math>does <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$a</mi></mrow></mfenced><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$a</mi></mrow></mfenced></mrow></math>&nbsp;or <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$a</mi></mrow></mfenced></mrow></mfenced></mrow></math>?@
qu.3.5.solution=@
qu.3.5.algorithm=$a=rint(-5,5);
condition:ne($a,0);
$m=maple("f:=abs(x-($a))/(x-($a)):
convert(f,string), MathML[ExportPresentation](f)");
$f=switch(0, $m);
$F=switch(1, $m);
$ANS=1;
$plot=plotmaple("plot($f,x=($a)-3..($a)+3,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.3.5.uid=44c233c4-c55f-42b2-aa57-9a71bb8c3717@
qu.3.5.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Intuitive Limits;
  Sub-Topic=One-sided limits;
  Difficulty=Easy;
@

qu.3.6.question=<p>Find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder></mrow></math>$F.</p>@
qu.3.6.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.3.6.allow2d=1@
qu.3.6.maple_answer=$ANS;@
qu.3.6.type=formula@
qu.3.6.mode=Maple@
qu.3.6.name=|x-a|/(x-a)LEFT@
qu.3.6.comment=<p>When <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$a</mi></mrow></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></mfenced></mrow></math>.</p>
<p>Here is a plot of the function.</p>
<p>$plot</p>@
qu.3.6.editing=useHTML@
qu.3.6.hint.1=When <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$a</mi></mrow></math>does <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/></mrow></math>or <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></mfenced></mrow></math>?@
qu.3.6.solution=@
qu.3.6.algorithm=$a=rint(-5,5);
condition:ne($a,0);
$m=maple("f:=abs(x-($a))/(x-($a)):
convert(f,string),MathML[ExportPresentation](f)
");
$f=switch(0,$m);
$F=switch(1,$m);
$ANS=-1;
$plot=plotmaple("plot($f,x=($a)-3..($a)+3,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.3.6.uid=955f86ec-ce52-49b7-b07b-09da9a562b83@
qu.3.6.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Intuitive Limits;
  Sub-Topic=One-sided limts;
  Difficulty=Easy;
@

qu.4.topic=LimitAtInfinity@

qu.4.1.question=<p>Find <math xmlns='http://www.w3.org/1998/Math/MathML'><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></math> <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true' mathsize='20'>&lpar;</mo></mrow></math>$Q<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true' mathsize='20'>&rpar;</mo></mrow></math>.</p>
<p>Enter infinity, -infinity, a finite number, or N if the limit does not exist.</p>@
qu.4.1.maple=evalb(($ANSWER)=($RESPONSE));@
qu.4.1.allow2d=1@
qu.4.1.maple_answer=limit(($T)/($B),x=infinity)@
qu.4.1.type=formula@
qu.4.1.mode=Maple@
qu.4.1.name=x->+infinity@
qu.4.1.comment=<p>Divide top and bottom by the highest power of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>in the denominator.</p>@
qu.4.1.editing=useHTML@
qu.4.1.solution=@
qu.4.1.algorithm=$a=rint(-5,5);
condition:ne($a,0);
$b=rint(-5,5);
$c=rint(-5,5);
$d=rint(-5,5);
condition:ne($d,0);
$e=rint(-5,5);
$f=rint(-5,5);
$g=rint(2);
$h=switch(rint(2),sin(x),cos(x));
$m=rint(3,10);
$n=rint(1,($m)-1);
$p=rint(3,10);
$q=rint(1,($p)-1);
$T="($a)*x^($m)+($b)*x^($n)+($g)*($h)+($c)";
$B="($d)*x^($p)+($e)*x^($q)+($f)";
$Q=maple("printf(MathML[ExportPresentation]((($T)/($B))))");

$plot=plotmaple("plot(($T)/($B),x=10..200,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.4.1.uid=e3bd0763-910d-4f8e-8d04-4d9e83bec026@
qu.4.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Intuitive Limits;
  Sub-Topic=Limits at infinity;
  Difficulty=Easy;
@

qu.4.2.question=<p>Find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></munder></mrow></math> <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true' mathsize='20'>&lpar;</mo></mrow></math>$Q<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true' mathsize='20'>&rpar;</mo></mrow></math>.</p>
<p>Enter infinity, -infinity, a finite number, or N if the limit does not exist.</p>@
qu.4.2.maple=evalb(($ANSWER)=($RESPONSE));@
qu.4.2.allow2d=1@
qu.4.2.maple_answer=limit($f,x=infinity)@
qu.4.2.type=formula@
qu.4.2.mode=Maple@
qu.4.2.name=sqrt(x^2+ax+b)-x@
qu.4.2.comment=<p>Multiple top and bottom by <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$a</mi></mrow></mfenced><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b</mi></mrow></mfenced></mrow></msqrt><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>x</mi></mrow></math>, then divide top and bottom by <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>(the highest power of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>in the denominator).</p>
<p>Here is a plot&nbsp; of the function</p>
<p>$plot</p>@
qu.4.2.editing=useHTML@
qu.4.2.hint.1=Rationalize the $place.@
qu.4.2.solution=@
qu.4.2.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
$z=rint(2);
condition:ne($a,$b);
$f=switch($z,"sqrt(x^2+($a)*x+($b))-x",
"1/(sqrt(x^2+($a)*x+($b))-x)");
$Q=maple("printf(MathML[ExportPresentation] ($f))");
$plot=plotmaple("plot($f,x=10..100,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");
$place=switch($z,"numerator", "denominator");@
qu.4.2.uid=48d676bf-200a-452b-a48b-585f18061633@
qu.4.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Intutitive Limits;
  Sub-Topic=Limits at infinity;
  Difficulty=Medium;
@

qu.4.3.question=<p>Find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></munder></mrow></math> <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true' mathsize='25'>&lpar;</mo></mrow></math>$Q<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true' mathsize='25'>&rpar;</mo></mrow></math>.</p>
<p>Enter infinity, -infinity, a finite number, or N if the limit does not exist.</p>@
qu.4.3.maple=evalb(($ANSWER)=($RESPONSE));@
qu.4.3.allow2d=1@
qu.4.3.maple_answer=limit($f,x=-infinity)@
qu.4.3.type=formula@
qu.4.3.mode=Maple@
qu.4.3.name=sqrt(x^2+ax+b)+x@
qu.4.3.comment=<p>Multiply top and bottom by <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$a</mi></mrow></mfenced><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b</mi></mrow></mfenced></mrow></msqrt><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow></math>, then divide top and bottom by&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>&nbsp;(the highest power of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>in the denominator). Remember when <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>is negative, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></math>.</p>
<p><br />
Here is a plot of the function.&nbsp;</p>
<p>$plot</p>@
qu.4.3.editing=useHTML@
qu.4.3.hint.1=Rationalize the $place.@
qu.4.3.hint.2=Recall, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0</mn></mrow></math>.@
qu.4.3.solution=@
qu.4.3.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
$z=rint(2);
condition:ne($a,$b);
$f=switch($z,"sqrt(x^2+($a)*x+($b))+x",
"1/(sqrt(x^2+($a)*x+($b))+x)");
$Q=maple("printf(MathML[ExportPresentation]($f))");
$plot=plotmaple("plot($f,x=-100..0,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");
$place=switch($z, "numerator", "denominator");@
qu.4.3.uid=253a2ec8-eb98-4c40-a8bc-ab9d15fac61d@
qu.4.3.info=  Course=Introduction to Calculus I;
  Author=Jack Weiner, Gord Clement;
  Difficulty=Hard;
  Topic=Intuitive Limits;
  Sub-Topic=Limits at infinity;
@

qu.4.4.question=<p>Find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mrow><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</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></munder></mrow></math>$Q.</p>
<p>Enter infinity, -infinity, a finite number, or N if the limit does not exist.</p>@
qu.4.4.maple=evalb(($ANSWER)=($RESPONSE));@
qu.4.4.allow2d=1@
qu.4.4.maple_answer=limit(($T)/($B),x=-infinity)@
qu.4.4.type=formula@
qu.4.4.mode=Maple@
qu.4.4.name=x->-infinity@
qu.4.4.comment=<p>Divide top and bottom by the highest power of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>in the denominator.</p>@
qu.4.4.editing=useHTML@
qu.4.4.solution=@
qu.4.4.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
$c=rint(-5,5);
$d=rint(-5,5);
$e=rint(-5,5);
$f=rint(-5,5);
condition:ne($a,0);
condition:ne($d,0);
$g=rint(2);
$h=switch(rint(2),sin(x),cos(x));
$m=rint(3,10);
$n=rint(1,($m)-1);
$p=rint(3,10);
$q=rint(1,($p)-1);
$T="($a)*x^($m)+($b)*x^($n)+($g)*($h)+($c)";
$B="($d)*x^($p)+($e)*x^($q)+($f)";
$Q=maple("printf(MathML[ExportPresentation]([($T)/($B)]))");

$plot=plotmaple("plot(($T)/($B),x=-200..0,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.4.4.uid=3479d5c5-1d0b-407f-87e4-5c2272146878@
qu.4.4.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Sub-Topic=Limits at infinity;
  Topic=Intuitive limits;
  Difficulty=Easy;
@

qu.5.topic=LimitsYieldingInfinity@

qu.5.1.question=<p>Find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><msup><mi mathvariant='normal'>$c</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder></mrow></math>$Q.</p>
<p>Enter infinity, -infinity, a finite number, or N (for None) if the limit does not exist.</p>@
qu.5.1.maple=evalb(($ANSWER)=($RESPONSE));@
qu.5.1.allow2d=1@
qu.5.1.maple_answer=limit($G,x=$c,right)@
qu.5.1.type=formula@
qu.5.1.mode=Maple@
qu.5.1.name=x->aRIGHT@
qu.5.1.comment=<p>Here is a plot of the function.</p>
<p>$plot</p>@
qu.5.1.editing=useHTML@
qu.5.1.solution=@
qu.5.1.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
$z=rint(3);
condition:ne($a,$b);
$n=rint(1,4);
$f=switch(rint(3),"1/(x-($a))^($n)","1/(($a)-x)^($n)",
"expand((x-($a))*(x-($b)))/(x-($a))^2");
$g=switch(rint(5),"cot(Pi*x)","cot(x)","csc(x)",
"csc(Pi*x)","ln(abs(x))");
$h=switch(rint(5),"tan(Pi*x/2)","tan(3*Pi*x/2)","sec(Pi*x/2)",
"sec(3*Pi*x/2)","ln(abs(x))");
$G=switch($z,"$f","$g","$h");
$Q=maple("printf(MathML[ExportPresentation]($G))");
$c=switch($z,$a,0,1);

$plot=plotmaple("plot($G,x=-abs($c)-2..abs($c)+2,y=-10..10,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.5.1.uid=9dd64412-fc16-4246-81b0-631132a08f5b@
qu.5.1.info=  Author=Jack Weiner, Gord Clement;
  Topic=Intuitive Limits;
  Sub-topic=One-sided limits;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
@

qu.5.2.question=<p>Find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi mathvariant='normal'>$c</mi></mrow></munder></mrow></math>$Q.</p>
<p>Enter infinity, -infinity, a finite number, or N (for None) if the left hand and right hand limits are different.</p>@
qu.5.2.maple=evalb(($ANSWER)=($RESPONSE));@
qu.5.2.allow2d=1@
qu.5.2.maple_answer=ans:=limit($F,x=$c);
if (ans<>undefined) then ans else N end if;@
qu.5.2.type=formula@
qu.5.2.mode=Maple@
qu.5.2.name=x->a@
qu.5.2.comment=<p>Here is a plot of the function.</p>
<p>$plot</p>@
qu.5.2.editing=useHTML@
qu.5.2.solution=@
qu.5.2.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
condition:ne($a,$b);
condition:ne($a,0);
$n=rint(1,4);
$z=rint(3);
$c=switch($z, $a, 0, 1);
$f=switch(rint(3),"1/(x-($a))^($n)","1/(($a)-x)^($n)",
"expand((x-($a))*(x-($b)))/(x-($a))^2");
$g=switch(rint(5),"cot(Pi*x)","cot(x)","csc(x)",
"csc(Pi*x)","ln(abs(x))");
$h=switch(rint(5),"tan(Pi*x/2)","tan(3*Pi*x/2)","sec(Pi*x/2)",
"sec(3*Pi*x/2)","ln(abs(x-1))");
$F=switch($z,"$f","$g","$h");
$Q=maple("printf(MathML[ExportPresentation]($F))");
$plot=plotmaple("plot($F,x=-abs($c)-2..abs($c)+2,y=-10..10,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.5.2.uid=b0751058-fb96-4c8f-bf63-4f8d207a5a95@
qu.5.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Intuitive limits;
  Sub-topic=Limt at a point;
  Difficulty=Easy;
@

qu.5.3.question=<p>Find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><msup><mi mathvariant='normal'>$c</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder></mrow></math>$Q.</p>
<p>&nbsp;</p>
<p>Enter infinity, -infinity, a finite number, or N (for None) if the limit does not exist.</p>@
qu.5.3.maple=evalb(($ANSWER)=($RESPONSE));@
qu.5.3.allow2d=1@
qu.5.3.maple_answer=limit($F,x=$c,left)@
qu.5.3.type=formula@
qu.5.3.mode=Maple@
qu.5.3.name=x->aLEFT@
qu.5.3.comment=<p>Here is a plot of the function.</p>
<p>$plot</p>@
qu.5.3.editing=useHTML@
qu.5.3.solution=@
qu.5.3.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
$z=rint(3);
condition:ne($a,$b);
$n=rint(1,4);
$f=switch(rint(3), "1/(x-($a))^($n)","1/(($a)-x)^($n)",
"expand((x-($a))*(x-($b)))/(x-($a))^2");
$g=switch(rint(5),"cot(Pi*x)","cot(x)","csc(x)",
"csc(Pi*x)","ln(abs(x))");
$h=switch(rint(5),"tan(Pi*x/2)","tan(3*Pi*x/2)","sec(Pi*x/2)",
"sec(3*Pi*x/2)","ln(abs(x))");
$F=switch($z, "$f","$g","$h");
$Q=maple("printf(MathML[ExportPresentation]($F))");
$c=switch($z, $a, 0, 1);

$plot=plotmaple("plot($F,x=-abs($c)-2..abs($c)+2,y=-10..10,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.5.3.uid=3e74c8fb-9a40-454c-a00c-465de06393e3@
qu.5.3.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Intuitive Limits;
  Sub-Topic=One-sided limits;
  Difficulty=Easy;
@

qu.6.topic=sin(at).sinbt)@

qu.6.1.question=<p>Evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mn>0</mn></mrow></munder></mrow></math>$F.</p>@
qu.6.1.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.6.1.allow2d=1@
qu.6.1.maple_answer=limit($f,x=0)@
qu.6.1.type=formula@
qu.6.1.mode=Maple@
qu.6.1.name=sin(ax)/sin(bx) etc@
qu.6.1.comment=<p>Recall, <math xmlns='http://www.w3.org/1998/Math/MathML'><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><mfrac><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mi>x</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></math>.</p>@
qu.6.1.editing=useHTML@
qu.6.1.hint.1=Recall <math xmlns='http://www.w3.org/1998/Math/MathML'><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><mfrac><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mi>x</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></math>.@
qu.6.1.solution=@
qu.6.1.algorithm=$a=rint(1,8);
$b=rint(1,8);
condition: ne($a,$b);
$f=switch(rint(5),"sin(($a)*x)/sin(($b)*x)",
"sin(($a)*x)/tan(($b)*x)","(1-cos(($a)*x))/x",
"(cos(($a)*x)-1)/x","tan(($a)*x)/tan(($b)*x)");
$F=maple("printf(MathML[ExportPresentation]($f))");@
qu.6.1.uid=a865f7ae-11b7-4320-8308-bb1345b2e18c@
qu.6.1.info=  Author=Jack Weiner, Gord Clement;
  Topic=Intuitive Limits;
  Sub-Topic=Trig Limits;
  Difficulty=Hard;
  Course=Introduction to Calculus I;
@

