Delta-Epsilon Definition of a Limit...3

Discussion in 'Calculus' started by nycmathguy, Apr 23, 2022.

  1. nycmathguy

    nycmathguy

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    Calculus
    Section 1.7

    Can you work out 31 in step by step fashion as a guide for me to do a few more on my own?

    Screenshot_20220422-230705_Samsung Notes.jpg
     
    nycmathguy, Apr 23, 2022
    #1
  2. nycmathguy

    MathLover1

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    31.

    Prove upload_2022-4-23_16-25-7.jpeg

    We need to manipulate the upload_2022-4-23_16-25-51.jpeg to show that |x+2|<something to set delta equal to that term:

    upload_2022-4-23_16-26-57.jpeg

    Since we cannot have a x term with epsilon, we let δ=1 and solve for the value x+2 would be:
    0<|x+2|<1

    −1<x+2<1

    −1−4<x+2−4<1−4

    −5<x−2<−3

    Here, we choose the larger value since if we chose the smaller value, the -5 would not be included, so:
    |x−2|<5

    Therefore,
    |x+2|<ε5

    Proof:

    upload_2022-4-23_16-29-20.jpeg
     
    MathLover1, Apr 23, 2022
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    nycmathguy likes this.
  3. nycmathguy

    nycmathguy

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    OMG! This is like Chinese backwards. Sorry but more time is needed to review this craziness.
     
    nycmathguy, Apr 24, 2022
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