qu.1.topic=Theory@

qu.1.1.mode=Multiple Choice@
qu.1.1.name=LimDefns(-inf,L)@
qu.1.1.comment=@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=@
qu.1.1.uid=962a4bbb-4594-49fd-a5dd-be0996978cc1@
qu.1.1.question=<p>For every <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&varepsilon;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn></mrow></math> there is an <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>N</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo><mn>0</mn></mrow></math>such that</p>
<p>when <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo><mi>N</mi></mrow></math> then <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>L</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo><mrow><mi>&epsiv;</mi></mrow></mrow></math>.</p>
<p>This is the definition for . . .</p>@
qu.1.1.answer=2@
qu.1.1.choice.1=<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='0.0em' rspace='0.0em'>&srarr;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'> </mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'> </mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></munder><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>L</mi></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math>@
qu.1.1.choice.2=<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><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&infin;</mi></mrow></munder><mo lspace='0.0em' rspace='0.0em'></mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>L</mi></mrow></math>@
qu.1.1.choice.3=<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>a</mi></mrow></munder><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.0em' rspace='0.0em'> </mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.1.choice.4=<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><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.1.choice.5=<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>a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><mo lspace='0.0em' rspace='0.0em'> </mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.1.choice.6=<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>a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo></mrow></msup><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.0em' rspace='0.0em'> </mo></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>L</mi></mrow></math>@
qu.1.1.choice.7=<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><msup><mi>a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo></mrow></msup></mrow></munder><mo lspace='0.0em' rspace='0.0em'></mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.1.fixed=@

qu.1.2.mode=Multiple Choice@
qu.1.2.name=LimDefns(-inf,inf)@
qu.1.2.comment=@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=@
qu.1.2.uid=2e2c0067-b649-4e75-b363-5d6c42313551@
qu.1.2.question=<p>For every <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>N</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn></mrow></math> there is an <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>M</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0</mn></mrow></math>such that when <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>M</mi></mrow></math> then <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;gt;</mo><mi>N</mi></mrow></math>.</p>
<p>This is the definition for . . .</p>@
qu.1.2.answer=4@
qu.1.2.choice.1=<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='0.0em' rspace='0.0em'>&srarr;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'> </mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'> </mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></munder><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>L</mi></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math>@
qu.1.2.choice.2=<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><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&infin;</mi></mrow></munder><mo lspace='0.0em' rspace='0.0em'></mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>L</mi></mrow></math>@
qu.1.2.choice.3=<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>a</mi></mrow></munder><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.0em' rspace='0.0em'> </mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.2.choice.4=<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><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.2.choice.5=<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>a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><mo lspace='0.0em' rspace='0.0em'> </mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.2.choice.6=<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>a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo></mrow></msup><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.0em' rspace='0.0em'> </mo></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>L</mi></mrow></math>@
qu.1.2.choice.7=<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><msup><mi>a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo></mrow></msup></mrow></munder><mo lspace='0.0em' rspace='0.0em'></mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.2.fixed=@

qu.1.3.mode=Non Permuting Multiple Choice@
qu.1.3.name=LimDefns(a+,L)@
qu.1.3.comment=@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=@
qu.1.3.uid=3ef590b9-91c3-4fbf-bee9-b4c447c68a73@
qu.1.3.question=<p>For every <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&varepsilon;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&amp;gt;</mo></mrow><mrow><mn>0</mn></mrow></math> there is a <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&delta;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn></mrow></math>such that when <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>0</mn><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>x</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>a</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>&delta;</mi></mrow></math> then <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='|' close='|' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'> </mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>L</mi><mo lspace='0.0em' rspace='0.0em'> </mo></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo></mrow><mrow><mi>&epsiv;</mi></mrow></math>.</p>
<p>This is the definition for . . .</p>@
qu.1.3.answer=6@
qu.1.3.choice.1=<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='0.0em' rspace='0.0em'>&srarr;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'> </mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'> </mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></munder><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>L</mi></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math>@
qu.1.3.choice.2=<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><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&infin;</mi></mrow></munder><mo lspace='0.0em' rspace='0.0em'></mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>L</mi></mrow></math>@
qu.1.3.choice.3=<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>a</mi></mrow></munder><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.0em' rspace='0.0em'> </mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.3.choice.4=<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><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.3.choice.5=<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>a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><mo lspace='0.0em' rspace='0.0em'> </mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.3.choice.6=<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>a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo></mrow></msup><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.0em' rspace='0.0em'> </mo></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>L</mi></mrow></math>@
qu.1.3.choice.7=<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><msup><mi>a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo></mrow></msup></mrow></munder><mo lspace='0.0em' rspace='0.0em'></mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.3.fixed=@

qu.1.4.mode=Multiple Choice@
qu.1.4.name=LimDefns(a-,-inf)@
qu.1.4.comment=@
qu.1.4.editing=useHTML@
qu.1.4.solution=@
qu.1.4.algorithm=@
qu.1.4.uid=094cc6b7-5c3e-4bf4-8e02-40a1cfba64b8@
qu.1.4.question=<p>For every <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>N</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo><mn>0</mn></mrow></math> there is a <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&delta;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn></mrow></math>such that when <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>0</mn><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'></mo><mi>a</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>&delta;</mi></mrow></math> then <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo><mi>N</mi></mrow></math>.</p>
<p>This is the definition for . . .</p>@
qu.1.4.answer=5@
qu.1.4.choice.1=<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='0.0em' rspace='0.0em'>&srarr;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'> </mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'> </mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></munder><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>L</mi></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math>@
qu.1.4.choice.2=<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><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&infin;</mi></mrow></munder><mo lspace='0.0em' rspace='0.0em'></mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>L</mi></mrow></math>@
qu.1.4.choice.3=<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>a</mi></mrow></munder><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.0em' rspace='0.0em'> </mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.4.choice.4=<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><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.4.choice.5=<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>a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><mo lspace='0.0em' rspace='0.0em'> </mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.4.choice.6=<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>a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo></mrow></msup><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.0em' rspace='0.0em'> </mo></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>L</mi></mrow></math>@
qu.1.4.choice.7=<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><msup><mi>a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo></mrow></msup></mrow></munder><mo lspace='0.0em' rspace='0.0em'></mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.4.fixed=@

qu.1.5.mode=Multiple Choice@
qu.1.5.name=LimDefns(a+,-inf)@
qu.1.5.comment=@
qu.1.5.editing=useHTML@
qu.1.5.solution=@
qu.1.5.algorithm=@
qu.1.5.uid=e38fd1f3-902b-4a97-a2a1-9e0498573a5e@
qu.1.5.question=<p>For every <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>N</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo></mrow><mrow><mn>0</mn></mrow></math> there is a <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&delta;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn></mrow></math>such that when <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>0</mn><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>x</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>a</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>&delta;</mi></mrow></math> then <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo><mi>N</mi></mrow></math>.</p>
<p>This is the definition for . . .</p>@
qu.1.5.answer=7@
qu.1.5.choice.1=<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='0.0em' rspace='0.0em'>&srarr;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'> </mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'> </mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></munder><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>L</mi></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math>@
qu.1.5.choice.2=<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><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&infin;</mi></mrow></munder><mo lspace='0.0em' rspace='0.0em'></mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>L</mi></mrow></math>@
qu.1.5.choice.3=<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>a</mi></mrow></munder><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.0em' rspace='0.0em'> </mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.5.choice.4=<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><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.5.choice.5=<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>a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><mo lspace='0.0em' rspace='0.0em'> </mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.5.choice.6=<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>a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo></mrow></msup><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.0em' rspace='0.0em'> </mo></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>L</mi></mrow></math>@
qu.1.5.choice.7=<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><msup><mi>a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo></mrow></msup></mrow></munder><mo lspace='0.0em' rspace='0.0em'></mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.5.fixed=@

qu.1.6.mode=Multiple Choice@
qu.1.6.name=LimDefns(a,inf)@
qu.1.6.comment=@
qu.1.6.editing=useHTML@
qu.1.6.solution=@
qu.1.6.algorithm=@
qu.1.6.uid=0aa8e112-8255-44d1-8164-262f0e54c524@
qu.1.6.question=<p>For every <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>N</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn></mrow></math> there is an <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mi>&delta;</mi></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn></mrow></math>such that when <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo></mrow><mrow><mfenced open='|' close='|' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>a</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow><mrow><mi>&delta;</mi></mrow></math> then <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;gt;</mo><mi>N</mi></mrow></math>.</p>
<p>This is the definition for . . .</p>@
qu.1.6.answer=3@
qu.1.6.choice.1=<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='0.0em' rspace='0.0em'>&srarr;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'> </mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'> </mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></munder><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>L</mi></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math>@
qu.1.6.choice.2=<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><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&infin;</mi></mrow></munder><mo lspace='0.0em' rspace='0.0em'></mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>L</mi></mrow></math>@
qu.1.6.choice.3=<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>a</mi></mrow></munder><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.0em' rspace='0.0em'> </mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.6.choice.4=<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><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.6.choice.5=<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>a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><mo lspace='0.0em' rspace='0.0em'> </mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.6.choice.6=<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>a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo></mrow></msup><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.0em' rspace='0.0em'> </mo></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>L</mi></mrow></math>@
qu.1.6.choice.7=<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><msup><mi>a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo></mrow></msup></mrow></munder><mo lspace='0.0em' rspace='0.0em'></mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.6.fixed=@

qu.1.7.mode=Multiple Choice@
qu.1.7.name=LimDefns(inf,L)@
qu.1.7.comment=@
qu.1.7.editing=useHTML@
qu.1.7.solution=@
qu.1.7.algorithm=@
qu.1.7.uid=42eb72af-e42c-479a-8d73-b7ace3daedd8@
qu.1.7.question=<p>For every <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&varepsilon;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn></mrow></math> there is an <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>N</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;gt;</mo><mn>0</mn></mrow></math>such that when <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;gt;</mo><mi>N</mi></mrow></math> then <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='|' close='|' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'> </mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>L</mi><mo lspace='0.0em' rspace='0.0em'> </mo></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mrow><mi>&epsiv;</mi></mrow></mrow></math>.</p>
<p>This is the definition for . . .</p>@
qu.1.7.answer=1@
qu.1.7.choice.1=<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='0.0em' rspace='0.0em'>&srarr;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'> </mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'> </mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></munder><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>L</mi></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math>@
qu.1.7.choice.2=<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><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&infin;</mi></mrow></munder><mo lspace='0.0em' rspace='0.0em'></mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>L</mi></mrow></math>@
qu.1.7.choice.3=<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>a</mi></mrow></munder><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.0em' rspace='0.0em'> </mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.7.choice.4=<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><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.7.choice.5=<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>a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><mo lspace='0.0em' rspace='0.0em'> </mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.7.choice.6=<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>a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo></mrow></msup><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.0em' rspace='0.0em'> </mo></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>L</mi></mrow></math>@
qu.1.7.choice.7=<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><msup><mi>a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo></mrow></msup></mrow></munder><mo lspace='0.0em' rspace='0.0em'></mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>@
qu.1.7.fixed=@

qu.1.8.mode=Multiple Choice@
qu.1.8.name=Epsilon@
qu.1.8.comment=@
qu.1.8.editing=useHTML@
qu.1.8.solution=@
qu.1.8.algorithm=@
qu.1.8.uid=b3927e4e-535e-4b92-b21a-e9406a6a199a@
qu.1.8.question=<p>In an "<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mi>&varepsilon;</mi><mrow><mi>&delta;</mi></mrow></mfrac></mrow></math>" limit question, the value of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&epsiv;</mi></mrow></math></p>@
qu.1.8.answer=1@
qu.1.8.choice.1=must be > 0@
qu.1.8.choice.2=may equal 0@
qu.1.8.choice.3=is sometimes <0@
qu.1.8.choice.4=is chosen first, and then we find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&delta;</mi></mrow></math>@
qu.1.8.choice.5=is found after we are given the value of delta@
qu.1.8.choice.6=can be less than 0 but only on Sundays@
qu.1.8.fixed=@

qu.1.9.mode=Multiple Selection@
qu.1.9.name=epsilon@
qu.1.9.comment=@
qu.1.9.editing=useHTML@
qu.1.9.solution=@
qu.1.9.algorithm=@
qu.1.9.uid=c1936416-69cc-4349-9a1e-7ab1dab1b24d@
qu.1.9.question=<p>In an epsilon (<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&epsiv;</mi></mrow></math>) delta (<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&delta;</mi></mrow></math>)&nbsp; limit question, the value of epsilon</p>@
qu.1.9.answer=1, 6@
qu.1.9.choice.1=must be > 0@
qu.1.9.choice.2=can equal 0@
qu.1.9.choice.3=can be < 0@
qu.1.9.choice.4=must be < 0@
qu.1.9.choice.5=is found after we are given the value of  delta@
qu.1.9.choice.6=is given first, and then we find a delta that works@
qu.1.9.fixed=@

qu.2.topic=E/DLinear@

qu.2.1.question=<p>In order to prove <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'>$c</mi></mrow></munder></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi></mrow></math>, we first let <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&varepsilon;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn mathvariant='normal'>0</mn></mrow></math> and then find the possible choices for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&delta;</mi></mrow></math> (ie., delta) so that if <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo></mrow></math>$X<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo><mrow><mi>&delta;</mi></mrow></mrow></math> then <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mrow><mi>&epsiv;</mi></mrow></mrow></math>.</p>
<p>&nbsp;</p>
<p>The BEST interval of solutions for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&delta;</mi></mrow></math> is . . . (Enter epsilon for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&epsiv;</mi></mrow></math>. Enter infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>.)</p>@
qu.2.1.maple=grade("$RESPONSE",$ANS);@
qu.2.1.allow2d=0@
qu.2.1.maple_answer=show($ANS)@
qu.2.1.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.2.1.type=maple@
qu.2.1.mode=Maple@
qu.2.1.name=lim(x->c)(b)=b@
qu.2.1.comment=<p>When the function is constant, any&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&delta;</mi></mrow></math> works!</p>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$b=rint(-5,5);
condition:ne($b,0);
$c=rint(-5,5);
$X=maple("printf(MathML[ExportPresentation](abs(x-($c))))");
$ANS='"(0,infinity)"';@
qu.2.1.uid=9b1597e3-f2fb-4de7-afaf-81f1b646faa0@
qu.2.1.info=  Author=Jack Weiner, Gord Clement;
  Topic=Formal Definition of a Limit;
  Sub-Topic=Find max delta;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
@

qu.2.2.question=<p><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'>$c</mi></mrow></munder></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&amp;lpar;</mo></mrow></math>$F <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$d</mi></mrow></math></p>
<p>In the plot below, the black line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$F. The red line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$d</mi></mrow></math>. The blue and green lines are respectively</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$d</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow><mrow><mi>&epsiv;</mi></mrow></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$d</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mi>&epsiv;</mi></mrow></math>. We want <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$d</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&epsiv;</mi></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow><mrow><mi mathvariant='normal'>$d</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow><mrow><mi>&epsiv;</mi></mrow></math>.</p>
<p>$plot</p>
<p>&nbsp;</p>
<p>In order to PROVE <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'>$c</mi></mrow></munder></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&amp;lpar;</mo></mrow></math>$F <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$d</mi></mrow></math>, we first let <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&varepsilon;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&amp;gt;</mo><mn mathvariant='italic'>0</mn></mrow></math> and then find the possible choices for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&delta;</mi></mrow></math> (ie., delta) so that</p>
<p>if <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo></mrow></math>$X<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo><mrow><mi>&delta;</mi></mrow></mrow></math> then <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&amp;verbar;</mo></mrow></math>$F<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow><mrow><mi mathvariant='normal'>$d</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow><mrow><mi>&epsiv;</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'></mo></mrow></math>.</p>
<p>The BEST interval of solutions for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&delta;</mi></mrow></math> is . . .</p>
<p>(Enter epsilon for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&epsiv;</mi></mrow></math>. And don't forget *. Example: for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&epsiv;</mi></mrow></mrow></math> enter 3/2*epsilon, not 3/2epsilon.)</p>@
qu.2.2.maple=grade("$RESPONSE",$ANS);@
qu.2.2.allow2d=0@
qu.2.2.maple_answer=show($ANS)@
qu.2.2.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.2.2.type=maple@
qu.2.2.mode=Maple@
qu.2.2.name=lim(x->c)(ax+b)@
qu.2.2.comment=<p>We want <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>y</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$d</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow><mrow><mi>&epsiv;</mi></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em'>&hArr;</mo></mrow><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow></math>$F<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$d</mi><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mrow><mi>&epsiv;</mi></mrow></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em'>&hArr;</mo></mrow></math>$sol1<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mrow><mi>&epsiv;</mi></mrow></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em'>&hArr;</mo></mrow></math>$sol2 $X <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mrow><mi>&epsiv;</mi></mrow></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em'>&hArr;</mo></mrow></math>$X<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow></math>$sol4</p>
<p>Therefore choose <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&delta;</mi></mrow></math>&nbsp;in <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></math>$sol4<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1111111em' stretchy='true'>&rsqb;</mo></mrow></math>.</p>@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=$a=rint(-6,6);
condition: ne($a,0);
$b=rint(-5,5);
condition:ne($b,0);
$c=rint(-3,3);
$m=maple("
convert(($a)/3*($c)+($b),string),
MathML[ExportPresentation](abs(x-($c))),
MathML[ExportPresentation](($a)/3*x+($b)),
MathML[ExportPresentation](abs(($a)/3*x+($b)-(($a)/3*($c)+($b)))),
MathML[ExportPresentation](epsilon/abs(($a)/3))
");
$d=switch(0,$m);
$X=switch(1,$m);
$F=switch(2,$m);
$sol1=switch(3,$m);
$sol4=switch(4,$m);
condition:lt(abs($d),8);
$ANS='"(0,epsilon/abs(($a)/3)]"';
$plot=plotmaple("plot([($a)/3*x+($b),$d,($d)+1,($d)-1],x=($c)-4..($c)+4,
view=[-10..10,-10..10],thickness=2,color=[black,red,blue,green]),plotdevice='gif', plotoptions='height=250,width=250'");
$sol2=mathml("abs(($a)/3)");@
qu.2.2.uid=b44d62fb-451e-4c0a-8ae8-fff21cc9d81c@
qu.2.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Formal Definition of a Limit;
  Sub-Topic=Find max delta;
  Difficulty=Easy;
@

qu.3.topic=lim(x^2-b)=a^2@

qu.3.1.mode=Multiple Choice@
qu.3.1.name=lim(x^2-b)=a^2,A neg@
qu.3.1.comment=@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$a=rint(-4,-1);
$b=rint(1,5);
$rs=$a^2-$b;
$X=mathml("x^2-$b");
$E=mathml("abs(x-$a)");
$n=abs(2*($a)-1);
$L=$n+1;
$S=$n-1;
$plot=plotmaple("plot([x^2-($b),$rs,($rs)+1,($rs)-1],x=($a)-2..($a)+2,
view=[-5..5,-7..16],thickness=2,color=[black,red,blue,green]),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.3.1.uid=bd334a52-7cb2-4304-88d6-e11c94f3f6e6@
qu.3.1.question=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi mathvariant='normal'>$a</mi></mrow></munder></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true' minsize='20px' maxsize='20px' >&lpar;</mo></mrow></math>$X<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'  minsize='20px' maxsize='20px'>&rpar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$rs</mi></mrow></math></p>
<p>In the plot below, the parabola (the black curve) is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$X. The red line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$rs</mi></mrow></math>. The blue and green lines  respectively are <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$rs</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow><mrow><mi>&epsiv;</mi></mrow></math>  and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$rs</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mi>&epsiv;</mi></mrow></math>. We want <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$rs</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&epsiv;</mi></mrow><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>y</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow><mrow><mi mathvariant='normal'>$rs</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow><mrow><mi>&epsiv;</mi></mrow></math>.</p>
<p>$plot</p>
<p>In proving&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi mathvariant='normal'>$a</mi></mrow></munder></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true' minsize='20px' maxsize='20px' >&lpar;</mo></mrow></math>$X<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true' minsize='20px' maxsize='20px' >&rpar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$rs</mi></mrow></math>, we let <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&varepsilon;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0.</mn></mrow></math> In order to find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&delta;</mi></mrow></math>, we will need a constraint on $E.  If we assume $E<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo><mn>1</mn></mrow></math>, the BEST interval of solutions for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&delta;</mi></mrow></math> is . . .</p>@
qu.3.1.answer=1@
qu.3.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=']' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>min</mi><mfenced open='{' close='}' separators=','><mrow><mn>1</mn><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mfrac><mi mathvariant='normal'>&varepsilon;</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></mfenced></mrow></mfenced></mrow></math>@
qu.3.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mi mathvariant='normal'>min</mi><mfenced open='{' close='}' separators=','><mrow><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mrow><mfrac><mi>&varepsilon;</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></mrow></mfenced></mrow></mfenced></mrow></math>@
qu.3.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mi mathvariant='normal'>min</mi><mfenced open='{' close='}' separators=','><mrow><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mrow><mfrac><mi>&varepsilon;</mi><mrow><mi mathvariant='normal'>$L</mi></mrow></mfrac></mrow></mrow></mfenced></mrow></mfenced></mrow></math>@
qu.3.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>min</mi><mfenced open='{' close='}' separators=','><mrow><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mfrac><mi>&varepsilon;</mi><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac></mrow></mrow></mfenced></mrow></mfenced></mrow></math>@
qu.3.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mi mathvariant='normal'>min</mi><mfenced open='{' close='}' separators=','><mrow><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mfrac><mi>&varepsilon;</mi><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac></mrow></mrow></mfenced></mrow></mfenced></mrow></math>@
qu.3.1.fixed=@

qu.3.2.mode=Multiple Choice@
qu.3.2.name=lim(x^2-b)=a^2,a pos@
qu.3.2.comment=@
qu.3.2.editing=useHTML@
qu.3.2.solution=@
qu.3.2.algorithm=$a=rint(1,4);
$b=rint(1,5);
$X=mathml("x^2-$b");
$rs=$a^2-$b;
$E=mathml("abs(x-$a)");
$n=(2*($a)+1);
$L=$n+1;
$S=$n-1;
$plot=plotmaple("plot([x^2-($b),$rs,($rs)+1,($rs)-1],x=($a)-2..($a)+2,
view=[-5..5,-7..16],thickness=2,color=[black,red,blue,green]),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.3.2.uid=1d3068cf-d8f3-439f-9dfb-9dbf9bb6d582@
qu.3.2.question=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi mathvariant='normal'>$a</mi></mrow></munder></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true' minsize='20px' maxsize='20px' >&lpar;</mo></mrow></math>$X<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true' minsize='20px' maxsize='20px' >&rpar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$rs</mi></mrow></math></p>
<p>In the plot below, the parabola (the black curve) is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$X. The red line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$rs</mi></mrow></math>. The blue and green lines  respectively are</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$rs</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow><mrow><mi>&epsiv;</mi></mrow></math>  and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$rs</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&epsiv;</mi></mrow></mrow></math>. We want <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$rs</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mrow><mi>&epsiv;</mi></mrow><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>y</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow><mrow><mi mathvariant='normal'>$rs</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mi>&epsiv;</mi></mrow></mrow></math>.</p>
<p>$plot</p>
<p>In proving&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi mathvariant='normal'>$a</mi></mrow></munder></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true' minsize='20px' maxsize='20px' >&lpar;</mo></mrow></math>$X<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true' minsize='20px' maxsize='20px' >&rpar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$rs</mi></mrow></math>, we let <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&varepsilon;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0.</mn></mrow></math> In order to find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&delta;</mi></mrow></math>, we will need a constraint on $E.  If we assume $E<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo><mn>1</mn></mrow></math>, the best interval of solutions for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&delta;</mi></mrow></math> is . . .</p>@
qu.3.2.answer=1@
qu.3.2.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=']' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>min</mi><mfenced open='{' close='}' separators=','><mrow><mn>1</mn><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mfrac><mi mathvariant='normal'>&varepsilon;</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></mfenced></mrow></mfenced></mrow></math>@
qu.3.2.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mi mathvariant='normal'>min</mi><mfenced open='{' close='}' separators=','><mrow><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mrow><mfrac><mi>&varepsilon;</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></mrow></mfenced></mrow></mfenced></mrow></math>@
qu.3.2.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=']' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>min</mi><mfenced open='{' close='}' separators=','><mrow><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mfrac><mi>&varepsilon;</mi><mrow><mi mathvariant='normal'>$L</mi></mrow></mfrac></mrow></mrow></mfenced></mrow></mfenced></mrow></math>@
qu.3.2.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>min</mi><mfenced open='{' close='}' separators=','><mrow><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mfrac><mi>&varepsilon;</mi><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac></mrow></mrow></mfenced></mrow></mfenced></mrow></math>@
qu.3.2.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mi mathvariant='normal'>min</mi><mfenced open='{' close='}' separators=','><mrow><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mfrac><mi>&varepsilon;</mi><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac></mrow></mrow></mfenced></mrow></mfenced></mrow></math>@
qu.3.2.fixed=@

qu.4.topic=lim(1/(x+b)=L@

qu.4.1.mode=Multiple Choice@
qu.4.1.name=lim(1/(x+b)=1/(b+c)@
qu.4.1.comment=@
qu.4.1.editing=useHTML@
qu.4.1.solution=@
qu.4.1.algorithm=$b=rint(1,5);
$c=rint(-2,2);
condition:gt(($c)+($b)-1,0);
$X=mathml("1/(x+$b)");
$m=maple("
MathML[ExportPresentation](abs(x-($c))),
convert(1/(($b)+($c)),string),
MathML[ExportPresentation](1/(($b)+($c))),
convert((($c)+($b)-1)*(($c)+($b)),string),
convert((($c)+($b)+1)*(($c)+($b)),string),
convert((($c)+($b))*(($c)+($b)),string),
MathML[ExportPresentation]((1/(($b)+($c)))+ epsilon),
MathML[ExportPresentation]((1/(($b)+($c)))- epsilon)
");
$xmc=switch(0,$m);
$l = switch(1,$m);
$L = switch(2, $m);
$C = switch(3,$m);
$D = switch(4,$m);
$E=switch(5,$m);
$lp=switch(6,$m);
$lm=switch(7,$m);
$plot=plotmaple("plot([1/(x+($b)),$l,($l)+.1,($l)-.1],x=($c)-3..($c)+3,
view=[-5..5,-1..1],thickness=2,color=[black,red,blue,green],discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.4.1.uid=507dd99e-5b85-42b2-8668-aa23a6f2cfda@
qu.4.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Formal Definition of a Limit;
  Difficulty=Medium;
  Sub-Topic=Limits with constraints;
@
qu.4.1.question=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi mathvariant='normal'>$c</mi></mrow></munder></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true' minsize='20px' maxsize='20px' >&lpar;</mo></mrow></math>$X<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><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math> $L</p>
<p>In the plot below, the black curve is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$X. The red line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$l</mi></mrow></math>. The blue and green lines respectively are <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$lp and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$lm. We want $lm<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo></mrow></math>$lp.</p>
<p>$plot</p>
<p>In proving <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi mathvariant='normal'>$c</mi></mrow></munder></mrow><mrow></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true' minsize='20px' maxsize='20px'>&lpar;</mo></mrow></math>$X<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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math> $L, we let <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&varepsilon;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn></mrow></math>. In order to find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&delta;</mi></mrow></math>, we will need a<strong> </strong>constraint<strong> </strong>on $xmc. If we assume $xmc<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo><mn>1</mn></mrow></math>, the BEST interval of solutions for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&delta;</mi></mrow></math> is . . .</p>@
qu.4.1.answer=1@
qu.4.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=']' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>min</mi><mfenced open='{' close='}' separators=','><mrow><mn>1</mn><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$C</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&varepsilon;</mi></mrow></mfenced></mrow></mfenced></mrow></math>@
qu.4.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>min</mi><mfenced open='{' close='}' separators=','><mrow><mn>1</mn><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$C</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&varepsilon;</mi></mrow></mfenced></mrow></mfenced></mrow></math>@
qu.4.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=']' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>min</mi><mfenced open='{' close='}' separators=','><mrow><mn>1</mn><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$D</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&varepsilon;</mi></mrow></mfenced></mrow></mfenced></mrow></math>@
qu.4.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>min</mi><mfenced open='{' close='}' separators=','><mrow><mn>1</mn><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$E</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&varepsilon;</mi></mrow></mfenced></mrow></mfenced></mrow></math>@
qu.4.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mi mathvariant='normal'>min</mi><mfenced open='{' close='}' separators=','><mrow><mn>1</mn><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$E</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&varepsilon;</mi></mrow></mfenced></mrow></mfenced></mrow></math>@
qu.4.1.fixed=@

qu.5.topic=lim(x->inf)(ax/(bx+c)=a/b@

qu.5.1.question=<p>The black curve is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$F. The red line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$A. The blue and green lines are respectively</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>$ap and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>$am. We want $am<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo></mrow></math>$ap.</p>
<p>$plot</p>
<p>&nbsp;</p>
<p>In order to PROVE <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mrow><mi>&infin;</mi></mrow></mrow></munder></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&amp;lpar;</mo></mrow></math>$F <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&amp;rpar;</mo></mrow></math>= $A we first let <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&varepsilon;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&amp;gt;</mo><mn mathvariant='italic'>0</mn></mrow></math> and then find the possible choices for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>N</mi></mrow></math> (a <strong>BIG POSITIVE NUMBER--</strong>we are heading to infinity, after all!<strong>) </strong>so that</p>
<p>if<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;gt;</mo><mi>N</mi></mrow></math> then <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&amp;verbar;</mo></mrow></math>$F<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow></math>$A<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&amp;rpar;</mo></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&amp;verbar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>&epsiv;</mi></mrow></math>.</p>
<p>The BEST interval of solutions for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>N</mi></mrow></math> is . . .</p>
<p>(Enter epsilon for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&epsiv;</mi></mrow></math> and infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>if you need them.)</p>@
qu.5.1.maple=grade("$RESPONSE",$ANS);@
qu.5.1.allow2d=0@
qu.5.1.maple_answer=show($ANS)@
qu.5.1.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.5.1.type=maple@
qu.5.1.mode=Maple@
qu.5.1.name=lim(x->inf)(a/(x+b))@
qu.5.1.comment=@
qu.5.1.editing=useHTML@
qu.5.1.solution=@
qu.5.1.algorithm=$a=rint(-5,5);
$A=mathml("$a");
condition: ne($a,0);
$b=rint(-5,5);
condition: ne($b,0);
$m=maple("
convert(abs(($a)*($b))/epsilon -($b),string),
MathML[ExportPresentation](($a)*x/(x+($b))),
MathML[ExportPresentation](($a)+ epsilon),
MathML[ExportPresentation](($a)- epsilon)
");
$N=switch(0,$m);
$F=switch(1,$m);
$ap=switch(2,$m);
$am=switch(3,$m);
$ANS='"[$N,infinity)"';
$plot=plotmaple("plot([($a)*x/(x+($b)),($a),($a)+1,($a)-1],x=10..100,thickness=2,color=[black,red,blue,green]),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.5.1.uid=89f63167-e666-4875-990d-d477453b8be9@
qu.5.1.info=  Author=Jack Weiner, Gord Clement;
  Topic=Formal Definition of a Limit;
  Sub-Topic=Limits at Infinity;
  Course=Introduction to Calculus I;
  Difficulty=Medium;
@

qu.6.topic=lim(a/(x-b))=inf@

qu.6.1.question=<p>In order to prove <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'>$b</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true' minsize='20px' maxsize='20px' >&lpar;</mo></mrow></math>$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><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>, we first let <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>N</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn mathvariant='italic'>0</mn></mrow></math> (Think of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>N</mi></mrow></math> as <strong>BIG!) </strong>and then find the possible choices for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&delta;</mi></mrow></math> (ie., delta) so that when <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo></mrow></math>$X<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo><mrow><mi>&delta;</mi></mrow></mrow></math>, we have $F<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;gt;</mo><mi>N</mi></mrow></math>. The BEST interval of solutions for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&delta;</mi></mrow></math> is . . .</p>
<p>(Note that we <strong>DON'T </strong>need absolute value on $X. Why not?)</p>@
qu.6.1.maple=grade("$RESPONSE",$ANS);@
qu.6.1.allow2d=0@
qu.6.1.maple_answer=show($ANS)@
qu.6.1.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.6.1.type=maple@
qu.6.1.mode=Maple@
qu.6.1.name=lim(x->b+)(a/(x-b))=inf@
qu.6.1.comment=<p>We don't need absolute value since the limit is from the right, so we know $X is positive.</p>@
qu.6.1.editing=useHTML@
qu.6.1.solution=@
qu.6.1.algorithm=$a=rint(1,12);
$b=rint(-5,5);
$f="($a)/(x-($b))";
$m=maple("
MathML[ExportPresentation](x-($b)),
MathML[ExportPresentation]($f)");
$X=switch(0,$m);
$F=switch(1,$m);
$ANS='"(0,$a/N]"';@
qu.6.1.uid=c70c967f-12a0-4a6f-83ad-a1e032f5381b@
qu.6.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Formal Definition of a Limit;
  Sub-Topic=Infinite Limit;
  Difficulty=Easy;
@

qu.6.2.question=<p>In order to prove <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'>$b</mi><mrow><mi mathvariant='normal'></mi></mrow></msup></mrow></munder></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&amp;lpar;</mo></mrow></math>$F <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&amp;rpar;</mo></mrow></math>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>, we first let <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>N</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn mathvariant='italic'>0</mn></mrow></math> (Think of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>N</mi></mrow></math> as BIG!)<strong> </strong>and then find the possible choices for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&delta;</mi></mrow></math> (ie., delta) so that when</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo></mrow></math>$X<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo><mrow><mi>&delta;</mi></mrow></mrow></math>, we have $F<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;gt;</mo><mi>N</mi></mrow></math>.</p>
<p>The MAXIMUM* value for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&delta;</mi></mrow></math> is . . .</p>
<p>*Note, in this question, we just want the maximum <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&delta;</mi></mrow></math>, not the interval <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=']' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>N</mi></mrow></mfenced></mrow></math>.</p>@
qu.6.2.maple=evalb(simplify(subs(N=757,($ANSWER)-($RESPONSE)))=0);@
qu.6.2.allow2d=0@
qu.6.2.maple_answer=$ANSWER@
qu.6.2.type=maple@
qu.6.2.mode=Maple@
qu.6.2.name=lim(x->b)(a/(x-b)^(2n))=inf@
qu.6.2.comment=@
qu.6.2.editing=useHTML@
qu.6.2.solution=@
qu.6.2.algorithm=$z=rint(5);
$a=switch($z,1,4,9,16,25);
$A=switch($z,1,2,3,4,5);
$b=rint(-5,5);
$f="($a)/(x-($b))^2";
$m=maple("
MathML[ExportPresentation](abs(x-($b))),
MathML[ExportPresentation]($f),
convert(($A)/N^(1/2),string)
");
$X=switch(0,$m);
$F=switch(1,$m);
$ANSWER=switch(2,$m);@
qu.6.2.uid=5180414f-cacd-47a6-99b1-90ff9e0db710@
qu.6.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Formal Definition of a Limit;
  Sub-Topic=Infinite Limits;
  Difficulty=Medium;
@

qu.6.3.question=<p>In order to prove <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'>$b</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&amp;lpar;</mo></mrow></math>$F <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mi>&infin;</mi></mrow></math>, we first let <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>N</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo><mn>0</mn></mrow></math> (Think of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>N</mi></mrow></math>as a big in magnitude negative number.)<strong> </strong>and then find the possible choices for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&delta;</mi></mrow></math> (ie., delta) so that when <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo></mrow></math>$X<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo><mrow><mi>&delta;</mi></mrow></mrow></math>, we have $F <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo><mi>N</mi></mrow></math>. The BEST interval of solutions for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&delta;</mi></mrow></math> is . . .</p>
<p>(Note that we don't need absolute value on $X but used $X instead of $Y. Why?)</p>@
qu.6.3.maple=grade("$RESPONSE",$ANS);@
qu.6.3.allow2d=0@
qu.6.3.maple_answer=show($ANS)@
qu.6.3.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.6.3.type=maple@
qu.6.3.mode=Maple@
qu.6.3.name=lim(x->b-)a/(x-b)=-inf@
qu.6.3.comment=<p>Don't be confused by the negative sign in the<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi mathvariant='normal'>$a</mi><mrow><mi>N</mi></mrow></mfrac></mrow></math>portion of the answer . Remember that <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>N</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo><mn>0</mn></mrow></math>so that<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi mathvariant='normal'>$a</mi><mrow><mi>N</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn></mrow></math>.</p>
<p>We don't need absolute value since the limit is from the left, therefore we know $X is positive.</p>@
qu.6.3.editing=useHTML@
qu.6.3.solution=@
qu.6.3.algorithm=$a=rint(1,12);
$b=rint(-5,5);
$m=maple("
MathML[ExportPresentation](($b)-x),
MathML[ExportPresentation](x-($b)),
MathML[ExportPresentation](($a)/(x-($b)))
");
$X=switch(0,$m);
$Y=switch(1,$m);
$F=switch(2,$m);
$ANS='"(0,-$a/N]"';@
qu.6.3.uid=40585561-2092-4a67-8e28-950ada5a9df8@
qu.6.3.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Medium;
  Topic=Formal Definition of a Limit;
  Sub-Topic=Infinite Limits;
@

