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Math Help - Continuous Extension...?

  1. #1
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    Continuous Extension...?

    Consider the function . How should f(x) be defined at to be continuous there? Give a formula for the continuous extension of f(x) that includes in its domain.

    F(x) = _____

    I haven't the slightest idea how to approach this. Do I just edit the original function somehow? This whole idea of changing a function to suit your own needs seems a bit sketchy to me. I've always been skeptical about doing what you want to something mathematic to make it work for you, as I'm paranoid it violates some sort of math rule and can't "really" be done. Either way... any tips on how to start this thing?
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  2. #2
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    Put the function as follows:

    f(x)=\left\{\begin{array}{cl}\dfrac{x^2-2}{x^4-5x^2+6}&\text{if }x\ne\sqrt2\\\alpha&\text{if }x=\sqrt2.\end{array}\right.

    We require that f(\sqrt2)=\lim_{x\to\sqrt2}\frac{x^2-2}{x^4-5x^2+6} to make that function continuous at x=\sqrt2, hence, compute the limit and you'll get the value of \alpha, and we're done.
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  3. #3
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    Thanks. So the <br /> <br />
\alpha,<br />
I'm looking for should be a constant? Because taking the limit of the original equation gives me -1. Am I supposed to do something else with that result? Because I'm under the impressions that the answer is supposed to be a formula...
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  4. #4
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    Yeah, \alpha is such value, then by finding it you'll satisfy continuity conditions and make the function continuous at that point.
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  5. #5
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    Well the assignment is online and it only accepts the correct answer. I tried submitting -1, but the program said it was wrong.

    Is there something else that I need to do? I'm still semi-convinced that my final answer should be an equation/formula.

    Ahhh. So it was may more simple than I thought.

    All I had to do was factor the polynomials and cancel and type in the new function that I got.

    But thanks for your help! :]
    Last edited by Krizalid; May 13th 2009 at 07:52 PM.
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