Complex Variables Limit Problem(s)

In summary: Just ∞. 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.
  • #1
wtmore
4
0

Homework Statement


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]


Homework Equations


?


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.
 
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  • #2
Try factoring the numerator and/or denominators. It's quite simple from there.
 
  • #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?
 
  • #4
wtmore said:
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?

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.
 
  • #5
SteveL27 said:
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.

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:
  • #6
wtmore said:
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?

Hmmm ... that didn't help. I'm stuck too now.
 
  • #7
SteveL27 said:
Hmmm ... that didn't help. I'm stuck too now.

The second one isn't indeterminant.
 
  • #8
Dick said:
The second one isn't indeterminant.

Would the answer be [itex]\pm∞[/itex]?
 
  • #9
wtmore said:
Would the answer be [itex]\pm∞[/itex]?

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
 
  • #10
wtmore said:
Would the answer be [itex]\pm∞[/itex]?

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

1. What is a complex variable limit problem?

A complex variable limit problem involves finding the limit of a complex-valued function as the input variable approaches a given complex number. It is similar to finding the limit of a real-valued function, but with the added complexity of working with complex numbers.

2. How do I solve a complex variable limit problem?

To solve a complex variable limit problem, you can use the same techniques as for real-valued limit problems. This includes algebraic manipulation, substitution, and using properties of limits such as the squeeze theorem. It is also important to understand the properties of complex numbers and how they behave in limit calculations.

3. What are some common challenges when solving complex variable limit problems?

Some common challenges when solving complex variable limit problems include working with complex numbers, which can be difficult to manipulate and visualize, and determining the behavior of the function near the limit point. It is also important to remember that the limit may not exist for certain functions, even if it exists for real-valued limit problems.

4. Can I use L'Hôpital's rule for complex variable limit problems?

No, L'Hôpital's rule only applies to real-valued functions. Instead, you can use the Cauchy-Riemann equations to calculate the limit of a complex variable function.

5. What are some real-world applications of complex variable limit problems?

Complex variable limit problems are used in many areas of science and engineering, including electromagnetism, fluid dynamics, and quantum mechanics. They are also used in signal processing and control systems to analyze and design complex-valued functions. These applications rely on the properties of complex numbers and their limits to model and predict real-world phenomena.

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