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anonymous

  • one year ago

For the limit lim x → 2 (x^3 − 4x + 3) = 3 illustrate the definition by finding the largest possible values of δ that correspond to ε = 0.2 and ε = 0.1. (Round your answers to four decimal places.) I know how to do it with a parabla but not cubic

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  1. IrishBoy123
    • one year ago
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    you need \(\delta\) such that \(|f(x) - 3| < \epsilon\) for \(0 \lt |x-2| \lt \delta\) it is a cubic but an easy one \(|x^3 − 4x + 3 - 3| \lt \epsilon\) \(|x^3 − 4x| \lt \epsilon\) \(|x(x-2)(x+2)| \lt \epsilon\) as for the rest: \(x = 2 \pm \delta\) here so i think we can say \(|2| \ |x-2| \ |4| \lt \epsilon\) \( |x-2| \lt \frac{\epsilon}{8}\) that might not be precise enough but you say you can do quadratics, so you might know better thus for \(0 \lt |x-2| \lt \delta\) we have \(0 \lt |x-2| \lt \frac{\epsilon}{8}\)

  2. anonymous
    • one year ago
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    Would you still need to subtract 2 since its subtract 2 in the middle?

  3. IrishBoy123
    • one year ago
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    if i understand you correctly, no. we are looking for a value for delta. i originally read your "I know how to do it with a parabla but not cubic" as suggesting the limits bit was fine, it was just playing with the cubic, but maybe i read wrong. we can say that for \(ε = 0.2 \ and \ 0.1\), \(\delta \lt 0.2/8 \ and \ \ 0.1/8\) are extremely good estimates [but i say estimates because the \(\frac{\epsilon}{8} \) is really \( \frac{\epsilon}{(2 \pm \delta)(4 \pm \delta) } \)] but they are asking for "the largest possible values of δ" which suggest that they want to push it to the limit???? and maybe a hand-wavey approach like this will be OK. i don't know. normally you might just assume that \(\epsilon\) is going to be small enough that \(\delta < 1\) is always true; so \(|x-2 | \lt 1\), so \(1\lt x\lt 3\) and thus we can play safe with \(\delta = \frac{\epsilon}{|x| \ |x+2|} \lt \frac{\epsilon}{3 \times 5}\) i don't know the background, but it occurs to me that you should also test it. once you have your \(\delta\), plug \(x = 2 \pm \delta\) into the equation and see if the test we are trying to pass, ie \( |f(x)−3|<ϵ\), is true for those values. then you can fine tune and make sure you are correct. this is one way:p http://www.wolframalpha.com/input/?i=multiply+%282%2B0.9999k%2F8%29%28%2B0.9999k%2F8%29%284%2B0.9999k%2F8%29 let me know what you think...

  4. IrishBoy123
    • one year ago
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    example 4 on this link, which starts half way down p3, better explains the usual kind of horse-trading or haircutting that you can do to justify limits in a delta epsilon proof, though in better detail that i have covered. *but* this is a long way away from finding the maximum value of \(\delta \)

  5. anonymous
    • one year ago
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    I think I understand it now, thanks

  6. anonymous
    • one year ago
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    Great work @IrishBoy123

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