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Complex Variables Limit Problem(s) 
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#1
Feb2612, 05:31 PM

P: 4

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
Feb2612, 05:33 PM

P: 29

Try factoring the numerator and/or denominators. It's quite simple from there.



#3
Feb2612, 05:42 PM

P: 4

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? 


#4
Feb2612, 05:52 PM

P: 800

Complex Variables Limit Problem(s)



#5
Feb2612, 06:04 PM

P: 4

[tex]\frac{z^2+i}{z^41}=\frac{z^2+i}{(z^21)(z^2+1)}=\frac{z^2+i}{(z1)(z+1)(zi)(z+i)}[/tex] Am I missing something in the numerator? EDIT: Would multiplying by the numerators conjugate be beneficial? 


#6
Feb2612, 06:17 PM

P: 800




#7
Feb2612, 06:36 PM

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#8
Feb2612, 06:41 PM

P: 4




#9
Feb2612, 06:44 PM

P: 800

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 


#10
Feb2612, 06:45 PM

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