Prove Limit Using Epsilon-delta Definition...4

Discussion in 'Calculus' started by nycmathguy, May 8, 2022.

  1. nycmathguy

    nycmathguy

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    Calculus
    Section 2.4

    I don't need to know this stuff to succeed in Calculus. However, I would like to have your math work reply file away as future study notes and/or reference notes. How about working out 20?

    Screenshot_20220507-174246_Samsung Notes.jpg
     
    nycmathguy, May 8, 2022
    #1
  2. nycmathguy

    MathLover1

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    lim((3/2)x-1/2, x->5)=7

    0<|x-5|<δ ; |((3/2)x-1/2)-7|<ε

    Step 1:

    observe this: ((3/2)x-1/2)-7|=|1/2 (3 x - 1)-7|

    Fix any ε>0. To have | (3 x - 1)/2-7| <ε

    | (3 x - 1)-14| <2ε
    |3x-15|<2ε......both side divide by 3
    |x-5|<2ε/3

    as ε>0

    0<x-5<2ε/3
    0<x<2ε/3+5

    Step 2:
    |x-5|<δ
    -δ <x-5<δ
    -δ +5<x<δ+5

    as δ>0
    0<x<δ+5

    Observe the relation between |x-c|<δ ; |(f(x))-L|<ε

    δ+5 =2ε/3+5
    δ =2ε/3

    Solution: δ =2ε/3
     
    MathLover1, May 9, 2022
    #2
    nycmathguy likes this.
  3. nycmathguy

    nycmathguy

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    You are amazingly gifted. You learned this stuff 47 years ago and never forgot what to do.

    How about the link below? No math threads until Friday.

    Prove Limit Using Epsilon-delta Definition...1
     
    nycmathguy, May 9, 2022
    #3
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