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Limit formal definition

  1. May 26, 2014 #1
    1. The problem statement, all variables and given/known data

    Use formal definition of limits
    Find L = lim x→ c f(x). Then find a number δ > 0 such for all x

    f(x) = 3 - 2x
    c = 3
    ε = 0.02

    3. The attempt at a solution

    limx→3 3 -2x

    limx→3 3 - limx→3 2x

    3 - 2(3) = -3
    L = -3



    I am not sure how to find delta
     
  2. jcsd
  3. May 26, 2014 #2

    LCKurtz

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    Ask yourself how close ##x## needs to be to ##3## so that ##|f(x)-L|<\epsilon## or, for your problem, ##|(3-2x) - (-3)|<.02##.
     
  4. May 26, 2014 #3

    lurflurf

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    |f(x)-L|<ε
    when
    |3-x|<δ
    write
    |f(x)-L|
    in terms of
    |3-x|
     
  5. May 26, 2014 #4
    ok so i take

    -0.02 < (3-2x)-(-3) < 0.02
    -0.02 < 6-2x < 0.02
    -6.02 < -2x < -5.98
    3.01 > x > -2.99
    (-2.99,3.01)

    -2.99 - 3 = -5.99
    3.01 - 3 = 0.1

    so since δ > 0
    δ = 0.1

    matches answer in book

    thank you for your help
     
  6. May 26, 2014 #5

    HallsofIvy

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    Notice that, at this point, you could say
    -0.02 < 2(3- x)< 0.02
    -0.01< 3- x< 0.01 so that |x- 3|< 0.01

     
  7. May 26, 2014 #6

    LCKurtz

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    That's good. But you could write it much neater:$$
    |6-2x| <.02$$ $$
    2|3-x| <.02$$ $$
    |3-x| <.01$$The steps are reversible so ##\delta=.01##.
     
  8. May 26, 2014 #7

    LCKurtz

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    Also note that should be -2.99 - (-3) = .01. And .1 should be .01.
     
    Last edited: May 26, 2014
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