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Math Help - Proving statement using the symbol definition of a limit

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    Proving statement using the symbol definition of a limit

    Prove that lim,x-->0 (x^2)=0 using ε,ζ.

    -I am unsure how to approach this.
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  2. #2
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    Re: Proving statement using the symbol definition of a limit

    Quote Originally Posted by johnsy123 View Post
    Prove that lim,x-->0 (x^2)=0 using ε,ζ.

    -I am unsure how to approach this.
    To show that \displaystyle \begin{align*} \lim_{x \to 0}x^2 = 0 \end{align*} you need to show \displaystyle \begin{align*} |x - 0| < \delta \implies \left|x^2 - 0\right| < \epsilon \end{align*}.

    \displaystyle \begin{align*} \left|x^2 - 0\right| &< \epsilon \\ \left|x^2\right| &< \epsilon \\ |x|^2 &< \epsilon \\ |x| &< \sqrt{\epsilon} \end{align*}

    So if we choose \displaystyle \begin{align*} \delta = \sqrt{\epsilon} \end{align*} we will be able to show this implication...

    \displaystyle \begin{align*} |x - 0| &< \delta \\ |x| &< \sqrt{\epsilon} \\ |x|^2 &< \epsilon \\ \left|x^2\right| &< \epsilon \\ \left|x^2 - 0\right| &< \epsilon \end{align*}

    Therefore \displaystyle \begin{align*} |x - 0| < \delta \implies \left|x^2 - 0\right| < \epsilon \end{align*}.

    Q.E.D.
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  3. #3
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    Re: Proving statement using the symbol definition of a limit

    Quote Originally Posted by johnsy123 View Post
    Prove that lim,x-->0 (x^2)=0 using ε,ζ.
    This is a clearly easy question. Look 0 < \varsigma  < 1 \Rightarrow 0 < {\varsigma ^2} < \varsigma  < 1
    So let \varepsilon  = \varsigma~!
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