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Math Help - prove limit by definition

  1. #1
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    prove limit by definition

    Hello everyone, this is my first topic on this forum, so if by chance this question was asked in the past, redirect me to the relevant post.

    So, I understand the definition of function limit, I just can't figure how to solve those problems.
    For example, I have
    lim(x->2)(x-4)=5

    So I write |x-9|<ε, 0<|x-2|<δ
    then
    |x-3||x+3|<ε
    and what's now? I know that probably I should assume that δ<1 or something, and then I stuck.
    If there a place when I can see just a big amount of examples, this would be great.

    Thank you.
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  2. #2
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    First of all, \displaystyle \lim_{x \to 2}(x^2 - 4) = 0, not \displaystyle 5.


    Now, to prove \displaystyle \lim_{x \to 2}(x^2 - 4) = 0, you need to show \displaystyle 0 < |x - 2| < \delta \implies |x^2 - 4 - 0| < \epsilon.


    So now we solve \displaystyle |x^2 - 4| < \epsilon for \displaystyle |x - 2|...


    \displaystyle |x^2 - 4| < \epsilon

    \displaystyle |x - 2||x + 2| < \epsilon

    \displaystyle M|x - 2| < \epsilon

    \displaystyle |x - 2| < \frac{\epsilon}{M}.


    We need to choose an \displaystyle M such that \displaystyle |x + 2| \leq M for \displaystyle x close to \displaystyle 2.

    If we require \displaystyle 0 < |x - 2| < 1, then

    \displaystyle x - 2 < 1

    \displaystyle x < 3

    \displaystyle x + 2 < 5

    \displaystyle |x + 2| < 5.

    So we choose \displaystyle M = 5, and thus we define \displaystyle \delta = \min \left \{ 1, \frac{\epsilon}{5}\right \}.


    Proof: Let \displaystyle \epsilon > 0 and define \displaystyle \delta = \min \left\{1, \frac{\epsilon}{5} \right\}.

    Then if \displaystyle 0 < |x - 2| < \delta we have


    \displaystyle |x^2 - 4| = |x - 2||x + 2|

    \displaystyle < 5|x - 2| since \displaystyle |x - 2| < \delta and \displaystyle \delta \leq 1

    \displaystyle < 5\left(\frac{\epsilon}{5}\right) since \displaystyle |x - 2| < \delta and \displaystyle \delta \leq \frac{\epsilon}{5}

    \displaystyle = \epsilon.


    Thus \displaystyle 0 < |x - 2| < \delta \implies |x^2 - 4 - 0| < \epsilon.
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  3. #3
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    Thanks! Now to check that I got it, I'll do another example, and you say if I'm right, OK?

    lim(x->0) (2x+3)/(x+2) = 3/2

    |x/(2x+4)|<ε

    we require that 0 < x < 1

    so M * x < ε

    x < ε/M
    and M > 1/(2x+4)

    -1 < x < 1
    -2 < 2x
    2 < 2x+4
    1/2 > 1/(2x+4)
    M = 1/2

    δ = min{1, 2ε}

    Right?

    BTW, I didn't find how to use Latex in order to write beautiful math expressions. The link in FAQ is broken.
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  4. #4
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    What you have done is fine.

    So now you need to do the proof, i.e. now show that if \displaystyle 0 < |x| < \delta then you can get \displaystyle \left|\frac{2x+3}{x+2} - \frac{3}{2}\right|<\epsilon.
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  5. #5
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    In my course I'm not required to provide the proof part. Maybe it's because we have a lot of material to learn and a small amount of time. I'm glad that I did it right, thank you for your explanation!
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