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Math Help - Prove Limit

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    Prove Limit

    Can someone show me how to solve this?

    Prove limit[x^3, x, 2]=8
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    Quote Originally Posted by sjenkins View Post
    Can someone show me how to solve this?

    Prove limit[x^3, x, 2]=8
    Don't CAS speak. Write it using correct mathematical notation:

    Prove \lim_{x \rightarrow 2} x^3 = 8.

    Are you required to do an epsilon-delta proof?
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    First of all, I have no idea what CAS speak is. Second, all the information I have is what I wrote. I am taking this as a correspondence course so I have no professor and write things the way I see them. I'm sorry if I'm doing something wrong. There are no calculus classes anywhere near me that I can take and this is my only way. This is a required class or I would not even be taking it in the first place.
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    Quote Originally Posted by sjenkins View Post
    Can someone show me how to solve this?

    Prove limit[x^3, x, 2]=8
    You want to show \lim_{x\to 2} x^3 = 8.

    Let |x-2| < \delta then it means ||x|-|2||\leq |x-2| < \delta \implies 2 - \delta< |x| < 2 + \delta.
    In addition let us require that 0 < \delta \leq 1 thus |x-2|<\delta \implies 1< |x| < 3.

    Now |x^3 - 8| = |(x-2)(x^2+x+4)| = |x-2||x^2+x+4| \leq |x-2|(|x|^2+|x|+4|) < |x-2|(9+3+4)

    Thus, we get that |x^3-8| < 16|x-2| < 16\delta.

    With this fact we can complete the proof. Let \epsilon > 0. And choose \delta = \min (1, \tfrac{\epsilon}{16} ).
    Then by the above reasoning if 0 < |x-2| < \delta = \tfrac{\epsilon}{16} then |x^3-8| < 16\delta \leq 16\cdot \tfrac{\epsilon}{16}= \epsilon.
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    Quote Originally Posted by sjenkins View Post
    First of all, I have no idea what CAS speak is. Second, all the information I have is what I wrote. I am taking this as a correspondence course so I have no professor and write things the way I see them. I'm sorry if I'm doing something wrong. There are no calculus classes anywhere near me that I can take and this is my only way. This is a required class or I would not even be taking it in the first place.
    You've said everything except the answer to the question I asked.

    Have you studied epsilon-delta proofs of limits?
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