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Math Help - Inequalities concerning limits

  1. #1
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    Inequalities concerning limits

    Hi you guys,

    so I have some difficulty understanding some of the content in my textbook. I recently started learning about limits and now I got to the formal defintion of a limit in my textbook which sounds/looks perfectly reasonable to me:

    For a function f defined in some open interval containing a (but not necessarily at a itself), we say
    \lim_{x \to a}f(x)=L,
    if given any number \epsilon > 0, there is another number \delta > 0, such that 0 < \left\vert x-a \right\vert < \delta guarantees that \left\vert f(x)-L \right\vert < \epsilon.

    Now this is shown in an example where it is to be proved that \lim_{x \to 2}x^2=4.
    Analagous to the definition it says there must be a \delta > 0 for which 0 < \left\vert x-2 \right\vert < \delta guarantees that \left\vert x^2-4 \right\vert < \epsilon for any given \epsilon.

    Next thing it says is the following:
    \left\vert x^2-4 \right\vert = \left\vert x+2 \right\vert \left\vert x-2 \right\vert,
    which makes sense since it's just factoring.

    Because our sole concern is what's in close proximity to x, it is assumed that x be in the interval \left [ 1,3 \right ] and from that it follows that \left\vert x+2 \right\vert \le 5.

    Now my actual problem:
    Why is (and that's what the textbook says) \left\vert x^2-4 \right\vert \le 5\left\vert x-2 \right \vert?

    That's probably some simple algebra and I'm overlooking something, but I really appreciate your help and thank you in advance.


    Greetings
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  2. #2
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    Re: Inequalities concerning limits

    Quote Originally Posted by Floele1106 View Post
    Now my actual problem:
    Why is (and that's what the textbook says) \left\vert x^2-4 \right\vert \le 5\left\vert x-2 \right \vert?
    We multiply both sides of the inequality |x + 2| ≤ 5 by |x - 2|. Since |x - 2| is nonnegative, the direction of the inequality does not change.
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  3. #3
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    Re: Inequalities concerning limits

    Well, that definitely makes sense. Thanks a lot.
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