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Complex Variables Limit Problem(s)

  1. Feb 26, 2012 #1
    1. The problem statement, all variables and given/known data
    a) [tex]\lim_{z\to 3i}\frac{z^2 + 9}{z - 3i}[/tex]
    b) [tex]\lim_{z\to i}\frac{z^2 + i}{z^4 - 1}[/tex]


    2. Relevant equations
    ????


    3. The attempt at a solution
    I'm assuming both of these are very, very similar, but I'm not quite sure how to solve them. I would like a method other than using ε and [itex]\delta[/itex].

    If you simply plug in the limit, it's obviously indeterminate. Is there an easy method to solve these limits or is the only option to use ε and [itex]\delta[/itex]? I'm not sure how to start, any suggestions would be helpful. Thanks.
     
  2. jcsd
  3. Feb 26, 2012 #2
    Try factoring the numerator and/or denominators. It's quite simple from there.
     
  4. Feb 26, 2012 #3
    Wow, can't believe I didn't realize that. It helped me solve a), which I ended up getting to be 6i, but b) cannot be factored (I don't think?).

    If it were [tex]z^4 + 1[/tex] in the denominator then I could, but I'm pretty sure I cannot factor anything in that problem?
     
  5. Feb 26, 2012 #4
    The denominator is a difference of squares. Then one of the factors has the same type of factorization as (a), namely the trick that a *sum* of squares can be factored with the use of an imaginary number.
     
  6. Feb 26, 2012 #5
    So I have:
    [tex]\frac{z^2+i}{z^4-1}=\frac{z^2+i}{(z^2-1)(z^2+1)}=\frac{z^2+i}{(z-1)(z+1)(z-i)(z+i)}[/tex]
    Am I missing something in the numerator?

    EDIT: Would multiplying by the numerators conjugate be beneficial?
     
    Last edited: Feb 26, 2012
  7. Feb 26, 2012 #6
    Hmmm ... that didn't help. I'm stuck too now.
     
  8. Feb 26, 2012 #7

    Dick

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    The second one isn't indeterminant.
     
  9. Feb 26, 2012 #8
    Would the answer be [itex]\pm∞[/itex]?
     
  10. Feb 26, 2012 #9
    Just [itex]∞[/itex]. There's only one point at infinity in the extended complex numbers. If you think of the plane, you go to infinity when you go toward the edge of the plane in any direction. There's only one complex infinity, way out there beyond the edge of the plane.

    A really nice visualization is to add a single point at infinity, and identify it with the "circumference" of the plane ... take the entire plane and fold it into a sphere, with the point at infinity at the north pole. It's called the Riemann sphere.

    http://en.wikipedia.org/wiki/Riemann_sphere
     
  11. Feb 26, 2012 #10

    Dick

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    Not in the complex numbers. It's a pole. Saying "does not exist" is probably safe.
     
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