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Basics of limits

  1. Jul 18, 2006 #1
    I know its a banal question but im new to calculus course and there are no math teachers in summer... also my book does not explain it

    Why it is
    [tex]\mid a_n - L \mid < \varepsilon [/tex]

    And not just
    [tex]L - a_n < \varepsilon [/tex]

    ??
    wouldn't the relation be the same?
     
    Last edited: Jul 18, 2006
  2. jcsd
  3. Jul 18, 2006 #2

    quasar987

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    If the definition of convergence did not include the absolute values, it would mean, for instance, that the constant sequence [itex]a_n = L+1[/itex] converges to L since for all n, [itex]L-(L+1)=-1<\epsilon[/itex].
     
  4. Jul 18, 2006 #3

    benorin

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    The statement [tex]\mid a_n - L \mid < \varepsilon [/tex] requires [tex]a_n[/tex] to be no more than [tex]\varepsilon [/tex] away from L, that is to say [tex]a_n\in (L- \varepsilon , L+ \varepsilon ) [/tex];
    whereas the statement [tex]L - a_n < \varepsilon [/tex] only requires [tex]a_n[/tex] less than L by no more than [tex]\varepsilon [/tex], that is to say [tex]a_n\in (L- \varepsilon , L) [/tex].
     
  5. Jul 18, 2006 #4

    Office_Shredder

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    Here's a more intuitive way of looking at it.. limits are all about showing two numbers get really close together...

    The best (only) way to measure distance in one dimension is absolute value. So of course you'd use that
     
  6. Jul 19, 2006 #5

    HallsofIvy

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    As office shredder said, |a- b| is the distance between numbers a and b. Of course, a distance is never negative and the distance from a to b is the same as the distance from b to a so it shouldn't matter which is larger. The number 7 is 3 steps away from the number 4. I could see that, of course by subtracting 7- 4= 3 but when I am using variables, I don't know which is larger. |7- 4|= |4- 7|= 3 regardless of which is larger.
     
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