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

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

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

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

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

    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) $\displaystyle \left\vert x^2-4 \right\vert \le 5\left\vert x-2 \right \vert$?
    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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