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Limit proof:but there is something wrong

  1. Feb 10, 2008 #1
    Hi everybody,

    Suppose one wants to prove that lim (3x²-x) = 10 as x approaches 2.

    Taking the definition of a limit we should have:
    "If 0 < |x-2| < d then it follows that |f(x) - 10| < e"

    Proving this statement, we can say |f(x)-10| < e
    iff |3x²-x-10| <e
    iff |3x+5|.|x-2| <e
    iff |x-2| < e/(|3X+5|)

    The problem that arises now is: how can one make the "x" in the denominator disappear?
    Because we need a delta in function of only an epsilon.

    So we could say: set d<=1 then it follows
    |x-2|<d<=1
    |x-2|<1
    -1<x-2<1
    1<x<3

    We know now that our x should lie between 1 and 3. But now comes the thing I fail to understand well: Why does one select 3 which is larger than the x should be. Why could we not just select 2 or 2.5 or something like that. So why should we have
    |x-2| < e/(|3.3+5|)
    |x-2| < e/(14)


    Finally we can set d=min{1, e/(14)} and we should have a solution for

    "If 0<|x-2|< d=e/(14) then |f(x) - 10|<e" and have proved that indeed
    lim (3x²-x) = 10 as x approaches 2.

    Could somebody tell me why I always should use that maximum border on x (i.c. 3) ? That's something that's been annoying me for about a week now!

    Thank you very much!
    M.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 10, 2008 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    You are taking the limit as x goes to 2. Further, since you have to take the limit "from both sides", x may well be larger than 3! The point is that x must, eventually, be "close" to 2 but it may be above or below 2. For x sufficiently close to 2, yes, it will be less than 3, but that is also true of any number you might pick. We can't select 2 because we could not be sure that x is "< 2". A number arbitrarily close to 2 might still be larger than 2.

    We certainly could use 2.5 - it's just that 1 tends to be easier to do arithmetic with than 1/2! If require that |x-2|< .5, then -.5< x- 2< .5. Adding 2 to both sides, 1.5< x< 2.5. Multiplying by 3, 4.5< 3x< 7.5. Adding 5, 9.5< 3x+ 5< 12.5. If 9.5< |3x+ 5|, then 1/|3x+5|< 1/9.5 so [itex]\epsilon/|3x+5|< \epsilon/9.5[/itex]. It's just easier to use 1 than 0.5.

     
    Last edited: Feb 11, 2008
  4. Feb 11, 2008 #3
    Thanks for the answer!
     
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