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Epsilon Delta Limit Definition

  1. Oct 16, 2012 #1
    1. The problem statement, all variables and given/known data

    Prove lim x--> -1

    1/(sqrt((x^2)+1)

    using epsilon, delta definition of a limit

    2. Relevant equations



    3. The attempt at a solution

    I know that the limit =(sqrt(2))/2

    And my proof is like this so far. Let epsilon >0 be given. We need to find delta>0 s.t. if 0<lx+1l<delta, then l[1/(sqrt((x^2)+1)]-(sqrt(2))/2l < epsilon. So we need to pick delta= ?

    I'm not sure how to arrive at the delta. I know I have to work out what's inside the absolute values, but I'm getting stuck.
     
  2. jcsd
  3. Oct 17, 2012 #2

    SammyS

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    What have you tried?

    Where are you stuck?
     
  4. Oct 17, 2012 #3
    I don't know how to simplify it at all. My thought was to maybe to get common denominator or and then multiply by the conjugate, but I don't know if this is correct. I got l[2-(2)^(1/2) *((x^2)+1)^(1/2)]/(2((x^2)+1)^(1/2)l. I'm sorry, it's hard to type it here, does any of that make sense?
     
  5. Oct 17, 2012 #4

    SammyS

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    Yes, that makes some sense.

    What you have is: [itex]\displaystyle \left|\frac{2-\sqrt{2}\sqrt{1+x^2}}{2\,\sqrt{1+x^2}}\right|[/itex] which is equivalent to: [itex]\displaystyle \left|\frac{1}{\sqrt{1+x^2}}-\frac{\sqrt{2}}{2}\right|[/itex]. That's a start.

    Rationalize the numerator. One factor of the result will be (x + 1) .

    Restrict δ to put a bound on the rest of the expression.
     
  6. Oct 17, 2012 #5
    When I try to rationalize, I get l(-2x^2)/[4((x^2)+1)^(1/2)+2(2)^(1/2)*((x^2)+1)]l

    What did I do wrong?
     
  7. Oct 17, 2012 #6

    SammyS

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    The numerator should be 2(1-x2) which is 1(1-x)(1+x).

    The denominator is a mess.
     
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