Prove Limit is 1/2

Discussion in 'Calculus' started by nycmathguy, May 12, 2022.

  1. nycmathguy

    nycmathguy

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    Calculus
    Section 2.4

    Can you work out 36 in full?
    Screenshot_20220510-193913_Samsung Notes.jpg
     
    nycmathguy, May 12, 2022
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  2. nycmathguy

    MathLover1

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    36.

    lim(1/x,x->2)=1/2

    Consider |f(x)-L|<ϵ

    |1/x-1/2|<ϵ

    |(2-x)/(2x)|<ϵ
    |2-x|<2xϵ
    |-(x-2|< -2xϵ
    -( -2xϵ)<|x-2|< -2xϵ
    2xϵ<|x-2|< -2xϵ

    since ϵ, choose -2xϵ
    |x-2|<δ

    then
    |x-2|<-2xϵ
    |x/(-2x)-2/(-2x)|<ϵ
    |-1/2+1/x|<ϵ
    |1/x-1/2|<ϵ ->proven
     
    MathLover1, May 12, 2022
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    nycmathguy likes this.
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  3. nycmathguy

    nycmathguy

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    Wow! Impressive.
     
    nycmathguy, May 12, 2022
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