New guy with a question about asymptotes

[138.veritas]

Hello everyone (not sure which forum to post this in, so MODS feel free to move it),

I've used this website as a reference many times in the past when I had a curious itch that needed to be scratched, so I'm hoping someone can soothe this particular nagging sensation once and for all.

Let me start by saying that I have not studied mathematics or physics beyond what was required for my psychology pre-requisites (pre-calculus and physics 1). That being said, a long lost fascination with asymptotes and displacement recently began consuming my waking thoughts again. Specifically, because asymptotes will forever near infinity without ever reaching it (let's just use f(x)=1/x for the sake of simplicity), is there any actual net difference in displacement? My conventional wisdom (generally wrong) would tell me no, there isn't. My rationale is that at any given point on the line, it is still no closer, nor farther away, from the forever expanding endpoint(s). This, to me, means that displacement (within the strict context of this example) does not exist.

Maybe displacement isn't the correct word to use here. Within the confines of a graph, different coordinates would indeed show a distinct linear "movement." But in relation to the overall line itself, there would be no positive or negative distinction, regardless of the coordinates. Granted, this is a completely two-dimensional approach.

My search skills haven't yielded any useful information about this specifically, so if anyone can chime in with a simple answer or supply a particular subject/book/website that can yield an answer I would greatly appreciate it.

Thanks,

-Dylan

 

[Bill Simpson]

Hundreds of years ago what we now call calculus was a method of solving problems that almost always worked, but people weren't exactly sure why. For simple "nice" things like lines and parabolas it always worked. Later complicated things that you probably wouldn't have thought of as functions were tried and errors were found.

Gradually all of calculus was reorganized in terms of limits and "epsilon and delta" proofs. (Later it became even more complicated, but I don't want to get you distracted and confused with that at this point.) This whole subject was given the name "analysis", and that is not "psychological analysis" since you mention psychology. Analysis is roughly as "the theory and proof that calculus works."

Since you are fascinated by this you might try to find a gentle introduction to analysis textbook. You could begin to work through that and try to discover how this stuff works.

Stillwell might be a good choice for an author with "Mathematics and its History." "Analysis by Its History" might be another good choice. You might be able to buy a reasonably priced copy of one of these, but I would recommend peeking at these in a university library before mailing your money off.

What I'm trying to do is come up with resources that will introduce you to the wonder of this subject and give you some feeling of how it developed, but not bury you in details that might diminish your fascination with the subject.

You can also Google for things like analysis calculus history and see if you can find good explanations, but a good book will probably be the next step after that.

I hope you have a good time

 

[Mark44]

Hello everyone (not sure which forum to post this in, so MODS feel free to move it),

Welcome to PF! As far as the right forum section, this is a good spot for your question.

I've used this website as a reference many times in the past when I had a curious itch that needed to be scratched, so I'm hoping someone can soothe this particular nagging sensation once and for all.

Let me start by saying that I have not studied mathematics or physics beyond what was required for my psychology pre-requisites (pre-calculus and physics 1). That being said, a long lost fascination with asymptotes and displacement recently began consuming my waking thoughts again. Specifically, because asymptotes will forever near infinity without ever reaching it (let's just use f(x)=1/x for the sake of simplicity), is there any actual net difference in displacement?

I don't understand your question, especially the "difference in displacement" part.

Let's look at your example function. The graph of f(x) = 1/x has two asymptotes: a horizontal asymptote (the line y = 0) and a vertical asymptote (the line x = 0).

For the horizontal asymptote, the function values approach 0 as x gets either very large or very negative. In calculus, we have notation to express these ideas; namely
$$\lim_{x \to \infty}\frac{1}{x} = 0$$
and something very similar as x approaches negative infinity.

If you look at specific numbers, f(100) = .01 and f(1000) = .001, so the larger x is, the closer f(x) is to zero. This might be related to what you're asking about difference in displacement, but I'm not sure.

For the vertical asymptote, as x gets close to zero, the function value gets either very large or very negative, depending on whether x is positive and near zero, or negative and near zero.

The notation for these ideas looks like this:
$$\lim_{x \to 0^+}\frac{1}{x} = \infty$$
$$\lim_{x \to 0^-}\frac{1}{x} = -\infty$$

Since the right- and left-side limits are different, the two-sided limit (as x approaches 0) doesn't exist.

Looking at some concrete numbers again, f(.01) = 100 and f(.001) = 1000. The closer x gets to zero (while remaining positive), the larger the function value gets.

The situation is similar for negative values of x. f(-.01) = -100 and f(-.001) = -1000. The closer x gets to zero (while remaining negative), the more negative the function value gets.

My conventional wisdom (generally wrong) would tell me no, there isn't. My rationale is that at any given point on the line, it is still no closer, nor farther away, from the forever expanding endpoint(s).

First off, the graphs of this and most other asymptotic functions are not lines. If you're referring to horizontal asymptotes, the farther out the x value is (away from 0), the closer the function value will be to the asymptote line.

"Forever expanding endpoint" makes no sense. The graphs of asymptotic functions don't have endpoints, let alone endpoints that are "forever expanding."

This, to me, means that displacement (within the strict context of this example) does not exist.

Maybe displacement isn't the correct word to use here. Within the confines of a graph, different coordinates would indeed show a distinct linear "movement." But in relation to the overall line itself, there would be no positive or negative distinction, regardless of the coordinates. Granted, this is a completely two-dimensional approach.

I don't understand any of this. If you can rephrase it, maybe I or someone else can answer what you're asking.

My search skills haven't yielded any useful information about this specifically, so if anyone can chime in with a simple answer or supply a particular subject/book/website that can yield an answer I would greatly appreciate it.

Thanks,

-Dylan

 

[138.veritas]

Thanks for the replies. I will look into those books Bill, and Mark, sorry for the confusion. I will try to be more clear using my limited mathematical vocabulary.

When I said line, I didn't mean to imply linear. I guess I should have said asymptotic curve? And when I said endpoints, it was more how I was visualizing it at the time... like Cartesian coordinates on a graph. I understand that by its very nature, an asymptote does not have an endpoint.

So, to try and clarify... if you have two reference points (correct terminology?) on a coordinate plane, you can figure out the difference (displacement?) in their positions in relation to one another. But if you compare each individual point to the asymptotic curve itself, how do you distinguish its place on a never ending curve (conceptually speaking, not the actual coordinate)? If you have a single point on a never ending curve, it is essentially always in the middle, correct? So if you have two points, what is the difference between the two points in relation to the curve (not to one another)?

I hope that is much clearer...

 

[Mark44]

Thanks for the replies. I will look into those books Bill, and Mark, sorry for the confusion. I will try to be more clear using my limited mathematical vocabulary.

When I said line, I didn't mean to imply linear.

A line is straight. Linear is an adjective that refers to straight-line behavior.

I guess I should have said asymptotic curve? And when I said endpoints, it was more how I was visualizing it at the time... like Cartesian coordinates on a graph. I understand that by its very nature, an asymptote does not have an endpoint.

So, to try and clarify... if you have two reference points (correct terminology?) on a coordinate plane, you can figure out the difference (displacement?) in their positions in relation to one another.

You can find their distance from one another, if that's what you're saying.

But if you compare each individual point to the asymptotic curve itself, how do you distinguish its place on a never ending curve (conceptually speaking, not the actual coordinate)? If you have a single point on a never ending curve, it is essentially always in the middle, correct?

Yes.

So if you have two points, what is the difference between the two points in relation to the curve (not to one another)?

Inasmuch as the distance between the two points would be finite, their distance relative to the overall curve (and expressed as a ratio) would be zero, since the length of the curve is infinite. I'm not sure how knowing this would be helpful, though.

Is there somewhere you're going with these questions?

I hope that is much clearer...