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Math Help - Definition of Limits

  1. #1
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    Definition of Limits

    So i know the definition (the epsilon, delta one)... i know how to use, but when i'm given an exercise which i have to factorice (sp?) like for example (x^2 - 9) which turns to |(x+3)(x-3)| then i have to do something else, and i dont really understand that part.

    Let's take this one

    lim as x-->2 (x^2 - 1) = 3

    Epsilon > 0 then there's a delta > 0 such that

    0 < |x - 2| < delta, then |(x^2 - 1 - 3| < epsilon

    0 < |x - 2| < delta, then |(x^2 - 4| < epsilon

    0 < |x - 2| < delta, then |x+2| |x-2| < epsilon

    Now the next step i know i have to pick a number, then do something i dont really understand.


    2. Also i'm having trouble with this one on the same subject

    f(x)=x^2 + 10x + 2, using the same delta epsilon argument to show that lim as x-->3 f(x)=f(3)

    Epsilon > 0 then there's a delta > 0 such that

    0 < |x - 3| < delta, then |x^2 + 10x + 2| < epsilon


    I cant even get the function to look like |x - 3| :/
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  2. #2
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    Quote Originally Posted by jikiami View Post
    So i know the definition (the epsilon, delta one)... i know how to use, but when i'm given an exercise which i have to factorice (sp?) like for example (x^2 - 9) which turns to |(x+3)(x-3)| then i have to do something else, and i dont really understand that part.

    Let's take this one

    lim as x-->2 (x^2 - 1) = 3
    For \epsilon > 0 choose \delta = \min \left\{ 1,\frac{\epsilon}{5} \right\}.

    Thus, if 0<|x-2|<\delta that means |x| \leq |x-2|+|2| < \delta + 2 thus, |x+2|\leq |x|+|2| < \delta + 4 \leq 5.
    Thus,
    |x^2-1-3|=|x^2-4| = |x-2||x+2| < 5\delta \leq \epsilon .
    Q.E.D.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    For \epsilon > 0 choose \delta = \min \left\{ 1,\frac{\epsilon}{5} \right\}.

    Thus, if 0<|x-2|<\delta that means |x| \leq |x-2|+|2| < \delta + 2 thus, |x+2|\leq |x|+|2| < \delta + 4 \leq 5.
    Thus,
    |x^2-1-3|=|x^2-4| = |x-2||x+2| < 5\delta \leq \epsilon .
    Q.E.D.
    Maybe you could explain me that as you would with someone living in the 3rd world? You did as to show the answer is right, but i just cant see it

    My doubts... i dont know why you picked the min. you did, where does the 4 and 5 come from... i just dont get it sorry
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