Precise Definition of Limits at Infinity...2

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

  1. nycmathguy

    nycmathguy

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    Calculus
    Section 2.6

    See Definition 9 below.

    Screenshot_20220530-090707_Samsung Notes.jpg

    Screenshot_20220530-094340_Samsung Notes.jpg
     
    nycmathguy, May 30, 2022
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  2. nycmathguy

    MathLover1

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    for the limit upload_2022-5-30_15-18-45.gif
    illustrate Definition 9 by finding a value of N that corresponds to M=100

    let function be defined on some interval (a, ∞), then lim(f(x))=∞ as x->∞means that for every positive number M there is corresponding positive number N such that if x>N then f(x)>M

    Definition 9:
    f(x) is continuous on an interval if it is continuous at every point of the interval

    Let M>0 and N=x*ln(x).

    Then, for all x>N, we have

    sqrt(x*ln(x))>sqrt(N)=sqrt(x*ln(x))=M.

    since given M=100, sqrt(x*ln(x))=100
    x ln(x) = 10000

    since N=x*ln(x), then N=10000

     
    MathLover1, May 30, 2022
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    nycmathguy likes this.
  3. nycmathguy

    nycmathguy

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    Wonderfully-done! I like it.
     
    nycmathguy, May 31, 2022
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