Existence of Limit of [(x+iy)/(x-iy)]^n

• kathrynag
In summary, the limit of [(x+iy)/(x-iy)]^n is a mathematical concept that represents the value that a function approaches as its input (x) approaches a certain value. It can be calculated using various methods such as algebraic manipulation, substitution, and L'Hopital's rule. However, the limit does not always exist and its existence is important in determining the behavior and properties of a function. It has practical applications in fields such as physics, engineering, and economics for modeling and analyzing real-world phenomena.
kathrynag

Homework Statement

For what values does the limit exist?
$$Lim_{n\rightarrow}\infty$$($$\frac{z}{z conjugate}$$)^n

The Attempt at a Solution

[(x+iy)/(x-iy)]^n
I just don't know how to tell when it exists. For even values of n because then there would be no sign changes?

Last edited:
It should be obvious that the existence or non-existence of the limit depends on z (alternatively, x and y) not on n, since n is the argument for the limit (aside: is there a name for that?)

What is the meaning of the limit of [(x+iy)/(x-iy)]^n?

The limit of [(x+iy)/(x-iy)]^n is a mathematical concept that represents the value that a function approaches as its input (x) approaches a certain value. In this case, the function is [(x+iy)/(x-iy)]^n and the input is x.

How is the limit of [(x+iy)/(x-iy)]^n calculated?

The limit of [(x+iy)/(x-iy)]^n can be calculated using various methods such as algebraic manipulation, substitution, and L'Hopital's rule. The specific method used depends on the complexity of the function and the value of n.

Does the limit of [(x+iy)/(x-iy)]^n always exist?

No, the limit of [(x+iy)/(x-iy)]^n does not always exist. It only exists if the function approaches a constant value (i.e. the function does not oscillate or approach infinity) as x approaches a certain value. If the function does not approach a constant value, the limit does not exist.

What is the significance of the existence of the limit of [(x+iy)/(x-iy)]^n?

The existence of the limit of [(x+iy)/(x-iy)]^n is important in determining the behavior and properties of a function. It helps in evaluating the continuity, differentiability, and convergence of the function, which are crucial in many mathematical and scientific applications.

How can the limit of [(x+iy)/(x-iy)]^n be used in practical applications?

The limit of [(x+iy)/(x-iy)]^n is used in various fields such as physics, engineering, and economics to model and analyze real-world phenomena. It also helps in solving problems involving rates of change, optimization, and approximation.

Replies
8
Views
1K
Replies
19
Views
5K
Replies
5
Views
1K
Replies
17
Views
1K
Replies
13
Views
2K
Replies
6
Views
777
Replies
1
Views
1K
Replies
20
Views
2K
Replies
44
Views
4K
Replies
9
Views
1K