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True/False regarding Delta Neighborhoods

  1. Mar 1, 2017 #1

    RJLiberator

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    Gold Member

    1. The problem statement, all variables and given/known data
    True/False: If a particular delta has been constructed as a suitable response to a particular epsilon challenge, then any smaller positive delta will also suffice.

    2. Relevant equations


    3. The attempt at a solution

    The submitted solution is as follows:
    mathhw1.png

    However, when I read this solution, I note that 0 < delta_1 < delta_2.

    The submitter goes on to start with delta_1 to then show that delta 2 holds. Didn't the submitter show that if a particular delta has been constructed as a suitable response to a particular epsilon challenge, then any LARGER positive delta will also suffice? This is not what the question asks.
     
  2. jcsd
  3. Mar 2, 2017 #2

    Mark44

    Staff: Mentor

    No, it is not saying that a larger delta works. Here's what I think is going on. A challenge value of ##\epsilon > 0## has been given, which is answered by a value of ##\delta_2##. In the image, a smaller value of ##\delta_1## is then selected. Now, if ##|x - c | < \delta_2## it will also be true (almost trivially) that ##|x - c | < \delta_1##, which in turn implies that ##|f(x) - f(c)| < \epsilon##
     
  4. Mar 2, 2017 #3

    PeroK

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    There are two aspects to limits: an understanding of what you are trying to do; and, the nitty-gritty manipulation of epsilons and deltas etc.

    In this case, the understanding should be clear: finding a delta means you are "close enough" to a point and if you reduce the delta you are "even closer". While, increasing the delta means you are "further away".

    You really shouldn't be having any trouble seeing this.
     
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