vertical asymptote or perforation

[Niqck]

Sorry if I use the wrong names, if you don’t understand what I am saying please let me know.

I have these two equations:
f(x) = (1 - sin^2(x)) / (cos(x))
and g(x) = (sin(x)) / (1 - cos^2(x))
I have to determine if these functions have a perforation or a vertical asymptote, I know both happen when the denominator = 0.
But what to do after it, i have no idea.
Could anyone help me with the next step?

 

[tkhunny]

Rule #1 - If there is a value that makes the denominator zero (0), then there is either a vertical asymptote or a hole (perforation).
Rule #2 - If the same value also makes the numerator zero (0), it is a hole.

Example: [math]y = \dfrac{x^{2}}{x}[/math] is EXACTLY the same as [math]y = x[/math] EXCEPT for x = 0.

 

[Dr.Peterson]

Rule #1 - If there is a value that makes the denominator zero (0), then there is either a vertical asymptote or a hole (perforation).
Rule #2 - If the same value also makes the numerator zero (0), it is a hole.

Example: [math]y = \dfrac{x^{2}}{x}[/math] is EXACTLY the same as [math]y = x[/math] EXCEPT for x = 0.

That's not strictly true; what about [MATH]y = \dfrac{x}{x^2}[/MATH]?

In general, you really have to think about limits.

I have these two equations:
f(x) = (1 - sin^2(x)) / (cos(x))
and g(x) = (sin(x)) / (1 - cos^2(x))
I have to determine if these functions have a perforation or a vertical asymptote, I know both happen when the denominator = 0.
But what to do after it, i have no idea.
Could anyone help me with the next step?

In these cases, simplifying will make it reasonably clear without explicitly talking about limits. Think about the Pythagorean identities, and show what you can do to simplify each function. That will make the behavior more visible.

I like your word "perforation". In America we tend to call a removable discontinuity a "hole", but that sounds so nontechnical!

 

[Niqck]

Rule #1 - If there is a value that makes the denominator zero (0), then there is either a vertical asymptote or a hole (perforation).
Rule #2 - If the same value also makes the numerator zero (0), it is a hole.

Example: [math]y = \dfrac{x^{2}}{x}[/math] is EXACTLY the same as [math]y = x[/math] EXCEPT for x = 0.

That's not strictly true; what about [MATH]y = \dfrac{x}{x^2}[/MATH]?

In general, you really have to think about limits.

In these cases, simplifying will make it reasonably clear without explicitly talking about limits. Think about the Pythagorean identities, and show what you can do to simplify each function. That will make the behavior more visible.

I like your word "perforation". In America we tend to call a removable discontinuity a "hole", but that sounds so nontechnical!

Thanks for the input, I’ll give it a try!

I just threw the term into google translate and it gave me perforation, I couldn’t easily find what the English term was supposed to be

 

[tkhunny]

That's not strictly true; what about [MATH]y = \dfrac{x}{x^2}[/MATH]?

I've been forgetting that for how many years? {shakes head}
Always assuming the degree of the denominator is less.
Probably take that error to my grave.

 

[Dr.Peterson]

I've been forgetting that for how many years? {shakes head}
Always assuming the degree of the denominator is less.
Probably take that error to my grave.

I've said it a few times myself, largely because I'm working from a textbook that is overly kind to the students by using only routine examples that avoid the worst cases. I (and the students) get too accustomed to what to expect in these problems, especially in a context where limits aren't in the vocabulary, so it's harder to give a full explanation. (You really have to talk about the multiplicity of individual factors, and so on, not just the overall degree ... and that's when it's a rational function.)

Unfortunately, here we're dealing with examples where this exact issue is part of the problem -- and it's definitely not just a routine rational function.