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Thread: Finding δ from a limt

  1. #1
    Junior Member
    Sep 2009

    Finding δ from a limt

    For the limit below, find values of δ that correspond to the ε values. (Round your answers to four decimal places.)

    For my answers i tried plugging into calculator and idk why i got them wrong:

    For ε = .5 i got .875
    for ε = .1 i got .975
    for ε =.05 i got .9875
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  2. #2
    Super Member
    Apr 2009
    Remember the definition of limit: For all $\displaystyle \epsilon >0$ there exists $\displaystyle \delta >0$ such that $\displaystyle \vert (4x-3)-1 \vert < \epsilon$ whenever $\displaystyle \vert x-1 \vert < \delta$ . Now work with arbitrary $\displaystyle \epsilon$, ie. $\displaystyle \vert 4x-4 \vert = 4 \vert x-1 \vert < \epsilon$ so for $\displaystyle \delta$ $\displaystyle \leq$ $\displaystyle \frac{ \epsilon }{4}$ we have $\displaystyle \vert x-1 \vert < \delta \leq \frac{ \epsilon }{4}$ and so $\displaystyle \vert (4x-3)-1\vert < \epsilon$ as required. And now you have your $\displaystyle \delta$ in terms of an arbitrary $\displaystyle \epsilon$ so you just have to plug in the values.
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