Specifying vertical asymptotes in periodic functions in set notation

In summary, the general set notation for specifying a vertical asymptote and domain for a periodic function is { x: x ∈ ℝ, x ≠ n⋅(π/2)+(π/4) }, where n is an integer. This notation correctly excludes the vertical asymptote from the domain and can be used for various values of n to exclude different points within the period. It is also recommended to use LaTeX for symbols in future posts.
  • #1
SubZer0
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Homework Statement
What is the general format for specifying recurring vertical asymptotes in periodic functions in set notation?
Relevant Equations
-2pi < x < 2pi
Hi all,

What is the general set notation for specifying a vertical asymptote and domain for a periodic function? For example, if I have a periodic function which has a period of pi/2, and within that period, a vertical asymptote occurs at pi/4. The domain is R, excluding that vertical asymptote. I am presuming that I can specify it something like:

{ x: x ∈ R, x ≠ n⋅(pi/2)+(pi/4) }, where n is an integer.

Does this look correct, or completely off?

Thanks!
 
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  • #2
SubZer0 said:
Hi all,

What is the general set notation for specifying a vertical asymptote and domain for a periodic function? For example, if I have a periodic function which has a period of pi/2, and within that period, a vertical asymptote occurs at pi/4. The domain is R, excluding that vertical asymptote. I am presuming that I can specify it something like:

{ x: x ∈ R, x ≠ n⋅(pi/2)+(pi/4) }, where n is an integer.

Does this look correct, or completely off?

Thanks!
Looks OK to me. An example of a function with this behavior is ##f(x) = \tan(2x)##
 
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  • #3
SubZer0 said:
Hi all,

What is the general set notation for specifying a vertical asymptote and domain for a periodic function? For example, if I have a periodic function which has a period of pi/2, and within that period, a vertical asymptote occurs at pi/4. The domain is R, excluding that vertical asymptote. I am presuming that I can specify it something like:

{ x: x ∈ R, x ≠ n⋅(pi/2)+(pi/4) }, where n is an integer.

Does this look correct, or completely off?

Thanks!
Let's check it out.
If n=0, then you are excluding π/4 from the domain. That's good.

If n=1, then you are excluding 3π/4 from the domain. That's good.

If n = −1, then you are excluding −π/4 from the domain. That's good.

Etc.

I'm curious about the inequality, −2pi < x < 2pi , that you have in the Relevant Equations .

Also, you can find many symbols by clicking on the icon 3rd from the right in the light blue banner at the top of the "Reply/Post thread" box.

242415

Using that, your result of
{ x: x ∈ R, x ≠ n⋅(pi/2)+(pi/4) }
becomes:
{ x: x ∈ ℝ, x ≠ n⋅(π/2)+(π/4) }

Even better, use LaTeX.
 
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  • #4
Thanks, Mark44 and SammyS for your responses. Have taken on board your advice for the symbols for future posts, SammyS.
 

1. What is a vertical asymptote in a periodic function?

A vertical asymptote in a periodic function is a vertical line that the graph of the function approaches but never touches. It represents a value that the function cannot take on because it would result in division by zero.

2. How do you specify vertical asymptotes in set notation?

To specify vertical asymptotes in set notation, you use the symbol ∞ (infinity) to represent the vertical line, followed by the value that the function approaches from either the left or right side. For example, if the function approaches a vertical asymptote at x = 2 from the left, it would be written as x → ∞ as x → 2-.

3. Can a periodic function have more than one vertical asymptote?

Yes, a periodic function can have multiple vertical asymptotes. This is because the function can approach different values from both the left and right side, resulting in multiple vertical lines that it cannot touch.

4. How do vertical asymptotes affect the behavior of a periodic function?

Vertical asymptotes can significantly impact the behavior of a periodic function, as they represent values that the function cannot take on. These values can create breaks or gaps in the graph of the function, and they can also affect the domain and range of the function.

5. Are there any other types of asymptotes in periodic functions?

Yes, in addition to vertical asymptotes, periodic functions can also have horizontal and slant asymptotes. These types of asymptotes represent values that the function approaches as x or y approaches infinity, respectively.

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