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Graphing Asymptotes

  1. Jul 24, 2012 #1
    1. The problem statement, all variables and given/known data
    Both the following is true for a particular choice of function f(x):

    For any ε>0 there exists an N>0 such that x>N --> |f(x) - 2| < ε
    For any ε>0 there exists an N<0 such that x<N --> |f(x)| < ε

    Sketch the graph of the function that satisfies both of these conditions.
    There are infinitely many correct answers, you need to only sketch the graph of one of them.
    Hint: can you say anything about the asymptotes of f(x)


    2. Relevant equations



    3. The attempt at a solution
    I'm not sure but can it be assumed the y-axis (y=0) and y = 2 are horizontal asymptotes? And thus any graph sketched that stays in it those boundaries is acceptable .
     
  2. jcsd
  3. Jul 24, 2012 #2

    ehild

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    Re: Graphing/Asymptotes

    The graphs need not stay between y=0 and y=2. Try to formulate the question in terms of limits. What are the limits of f(x) at +infinity and at -infinity?

    ehild
     
  4. Jul 24, 2012 #3
    Re: Graphing/Asymptotes

    Hmmm... I would think it would be positive infinity and negative infinity respectively as limits, no?
     
  5. Jul 24, 2012 #4

    HallsofIvy

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    Re: Graphing/Asymptotes

    No. "for x> N", in other words, for x very large, [itex]|f(x)-2|<\epsilon[/itex] So what is f(x) close to for x very large?
     
  6. Jul 24, 2012 #5
    Re: Graphing/Asymptotes

    The only thing I see from that is that f(x) is within epsilon units of 2? But what should I gather from that?
     
  7. Jul 24, 2012 #6

    HallsofIvy

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    Re: Graphing/Asymptotes

    Since that is true for any positive epsilon, you should gather that f(x) is very close to 2! And getting closer to 2 as x gets larger.
     
  8. Jul 24, 2012 #7
    Re: Graphing/Asymptotes

    Ahh okay, that makes sense lim f(x) approaches 2 as x tends to ∞. But what does the other limit tell us, that when x approaches negative infinity, f(x) approaches 0?
     
  9. Jul 24, 2012 #8

    Mark44

    Staff: Mentor

    Re: Graphing/Asymptotes

    Yes.
     
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