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Midterm

2005 Test 1 solution

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Department
Mathematics
Course
MAT137Y1
Professor
N/ A
Semester
Winter

Description
MAT137Y20052006TermTest1Solutions1310aSolveforallxwhichsatises2xx12Bydenitionofabsolutevaluexx0x1x1xx1xx0x1x153Ifx1then2xx12xx13x1whenxbutthisisnota26validsolutionsinceweareonlylookingatvaluesx113xwhichisavalidIf1x0then2xx12xx1x122solutionsincex1031Ifx0then2xx12xx13x1xwhichisalsoavalidsolution2611Hencetheequationissatisedforx26xx210bSolvetheinequalityandexpressyouranswerasaunionofintervalsx1x5Notethattheinequalitydoesnotmakesenseforx15Movingeverythingtoonesidex2xx2xx5x2x1x00x1x5x1x5x1x56x20x1x5Wecanuseachartofsignstoindicatewhereeachofthefactorsispositiveornegative11x55x11xx336x2x1x56x2x1x51Sobythechartweseethattheinequalityissatisedforx51325aGivetheprecisedenitionofthefollowingstatementlimfxLxaWesaythatlimfxLifforany0thereexists0suchthat0xaxafxL35x9directlyusingtheprecisedenitionoflimit13bProvethatlim2x3x1412Choosemin1Suppose0x3Then23225x945x93x13x20x333x113x32222x144x14x14x13x11x324x11
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