Precisel Definition of Limit at Infinity

Discussion in 'Calculus' started by nycmathguy, May 30, 2022.

  1. nycmathguy

    nycmathguy

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    Calculus
    Section 2.6

    See Definition 7 below.

    Screenshot_20220530-090652_Samsung Notes.jpg

    Screenshot_20220530-093906_Samsung Notes.jpg
     
    nycmathguy, May 30, 2022
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  2. nycmathguy

    MathLover1

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    upload_2022-5-30_15-42-51.gif


    Given ϵ =0.1 and ϵ=0.05

    if x>N then |f(x)-L|<ϵ


    |(1-3x)/sqrt(x^2+1)-(-3)|<0.1

    |(1-3x)/sqrt(x^2+1)+3|<0.1

    x>11.2826 => integer solutions x>12

    |(1-3x)/sqrt(x^2+1)-(-3)|<0.05

    |(1-3x)/sqrt(x^2+1)+3|<0.05

    x>21.3791 => integer solutions x>22


    since x>N, fist integer solution is N=11
     
    MathLover1, May 30, 2022
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    nycmathguy likes this.
  3. nycmathguy

    nycmathguy

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    Thank you. Trust me, I will seriously study all your replies to my threads involving proofs when time allows.
     
    nycmathguy, May 31, 2022
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