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Math Help - Limit Definition

  1. #1
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    Limit Definition

    Need help proving something.
    Show using the definition of a limit that lim (as x approaches c) of (x^3)=(c^3) for all c in R.
    Thank you in anticipation.
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  2. #2
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    Quote Originally Posted by Fares23 View Post
    Need help proving something.
    Show using the definition of a limit that lim (as x approaches c) of (x^3)=(c^3) for all c in R.
    Thank you in anticipation.
    If c\geq 0 then,
    For every \epsilon>0
    There is \delta>0
    such as,
    if |x-c|<\delta is in domain,
    then,
    |x^3-c^3|<\epsilon.

    The first inequality can be expressed as,
    c-\delta<x<c+\delta
    And,
    c^3-\epsilon<x^3<c^3+\epsilon
    Thus,
    \sqrt[3]{c^3-\epsilon}<\sqrt[3]{-\epsilon}<x<\sqrt[3]{\epsilon}<\sqrt[3]{c^3+\epsilon}
    If you chose,
    c+\delta=\sqrt[3]{\epsilon}
    c-\delta=\sqrt[3]{-\epsilon}
    Thus,
    \delta=-c+\sqrt[3]{\epsilon}
    Would make the inequality work can gaurentte the function is defined.

    Now you do the second half if, c<0.

    There got to be a more simple way to do it. But that is the only one that I came up with.
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  3. #3
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    You don't have to write out the whole thing, but what changes for c<0?
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  4. #4
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    Quote Originally Posted by Fares23 View Post
    You don't have to write out the whole thing, but what changes for c<0?
    After manipulating the epsilon I got,
    \sqrt[3]{c^3-\epsilon}<x<\sqrt[3]{c^3+\epsilon}
    But, since c> 0
    \sqrt[3]{\epsilon}<\sqrt[3]{c^3+\epsilon}
    That was what I was trying to do.

    But the problem is I just realized that what I did was wrong. Maybe I will try to re-edit what I posted later on.
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  5. #5
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    What did you do wrong in the first part?
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  6. #6
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    Quote Originally Posted by Fares23 View Post
    What did you do wrong in the first part?
    The inequality,
    \sqrt[3]{c^3-\epsilon}<\sqrt[3]{\epsilon}
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