Proving the Limit of (z/\bar{z})2 Does Not Exist

  • Thread starter Rubik
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In summary, to show that the limit does not exist, we can take the limit approaching 0 along the real axis and along the imaginary axis and show that the results are different. This is done by substituting z=x+iy and taking the limit at x=0. The limit along the real axis is 1, while the limit along the imaginary axis is -1, proving that the limit does not exist.
  • #1
Rubik
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Homework Statement



Show that the lim z→0 of (z/[itex]\bar{z}[/itex])2 does not exist

Homework Equations





The Attempt at a Solution


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  • #2
Write it out with z=x+iy.
 
  • #3
Take the limit approaching 0 along the real axis and along the imaginary axis. Show that the results are different.
 
  • #4
I have come up with this

Taking the limit along the Real axis:
lim as z→0 of (z/[itex]\bar{z}[/itex])2
= lim (x + 0i)2/(x - 0i)2
= lim x2/x2
= 1

Then taking the limit at the points x + xi for x→0:
lim as z→0 of (z/[itex]\bar{z}[/itex])2
= lim (x + xi)2/(x - xi)2
= lim (2x2)/(-2x2)
= -1

and since 1 ≠ -1 The limit does not exist.
 

1. What is the limit of (z/\bar{z})2 as z approaches 0?

The limit of (z/\bar{z})2 as z approaches 0 does not exist. This means that as z gets closer and closer to 0, the function does not approach a specific value.

2. How do you prove that the limit of (z/\bar{z})2 does not exist?

To prove that the limit of (z/\bar{z})2 does not exist, you would need to show that for every possible value that the function could approach, there is a point where the function does not approach that value. This can be done through various methods, such as using the epsilon-delta definition of a limit or using graphical representations.

3. Can you give an example of a point where the limit of (z/\bar{z})2 does not exist?

One example of a point where the limit of (z/\bar{z})2 does not exist is z=0. At this point, the function is undefined and does not approach any specific value as z approaches 0.

4. Is the limit of (z/\bar{z})2 always undefined?

No, the limit of (z/\bar{z})2 may exist at certain points. It is only undefined at points where the function is not defined or where it does not approach a specific value.

5. Is this function commonly used in real-world applications?

The function (z/\bar{z})2 may be used in certain mathematical models, but it is not commonly used in real-world applications. It is more commonly used in theoretical mathematics and in understanding the concept of limits.

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