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Math Help - Solving a delta epsilon proof given a graph

  1. #1
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    Solving a delta epsilon proof given a graph

    My professor started covering these today and i have never seen them before, even though I took a high school calculus class. How do I go about solving this?

    (Problem attached)
    Attached Thumbnails Attached Thumbnails Solving a delta epsilon proof given a graph-proof-1.jpg  
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  2. #2
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    You can read the first part of the condition as,

    "if the distance of x from 5 is less than delta",

    and the second part as,

    "then the distance of y from 3 is less than 0.6".
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  3. #3
    MHF Contributor

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    This problem is actually much easier than if you are given a formula for the function and you really need to understand it.

    "3.6" and "2.4" are 3+ .6 and 3- .6 so the two horizontal red lines on your graph show the boundaries for " |f(x)- 3|< .6". In order that f(x) be within those two lines x must be within the two vertical red lines. Those are at 4 and 5.7 which are different distances from 5: |4- 5|= 1 and |5.7- 5|= .7. But since .7< 1, if |x- 5|< .7 certainly |x- 5|< 1 will also be true. So we stay within the two vertical red lines, and so the function value is with in the two horizontal red lines as long as |x- 5|< .7.
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