A question about some concepts from solving limits

[xwolfhunter]

Simple question, simple example.
$$(x-1)$$
$$\frac{(x-2)(x-1)}{x-2}$$
So, it's quite clear that the contents of the two lines are not equal to one another. Since $\frac{1}{1}\neq\frac{x-2}{x-2}$, how is it the case that we can treat it as $\frac{1}{1}=\frac{x-2}{x-2}$ when we're doing limits? Edit: Oh wait, that's the leap of faith used all throughout calculus, isn't it? My b. I'd like to see a proof though, where could I find one?

And more generally, is there a field of mathematics that treats this kind of "almost equal but not quite" trickiness? Where is the study of the almost equal? Please someone ease my cramped eyebrows. I know that calculus makes use of this kind of thing at every possible opportunity, but is there a field of study that distinctly and separately treats those concepts? The study of the infinitesimal yet tangible.

 

[S.G. Janssens]

And more generally, is there a field of mathematics that treats this kind of "almost equal but not quite" trickiness?

(Mathematical) analysis.

See for example https://www.physicsforums.com/threads/list-of-mathematics-books.668404/ for some pointers to the literature.

 

[axmls]

This is directly as a result of the fact that when taking a limit, we don't care what actually happens at the point (only what happens very close to it).

 

[Mark44]

Simple question, simple example.
$$(x-1)$$
$$\frac{(x-2)(x-1)}{x-2}$$
So, it's quite clear that the contents of the two lines are not equal to one another. Since $\frac{1}{1}\neq\frac{x-2}{x-2}$, how is it the case that we can treat it as $\frac{1}{1}=\frac{x-2}{x-2}$ when we're doing limits? Edit: Oh wait, that's the leap of faith used all throughout calculus, isn't it? My b. I'd like to see a proof though, where could I find one?

Except at x = 1, the graphs of y = x - 1 and $y = \frac{(x - 1)(x - 2)}{x - 2}$ are exactly the same. The only difference between the two graphs is that the graph of the latter equation has a removable discontinuity -- otherwise known as a "hole" -- at (2, 1).

The expression $\frac{x - 2}{x -2}$ is undefined for only one value: x = 2. As is usually the case where functions become undefined, we see if the limit exist as x gets arbitrarily close to 2. In this case, since $\frac{x - 2}{x -2}$ remains close to 1 for any choice of x other than 2, it seems fairly obvious that the limit should be 1. That assertion can be verified rigorously using the $\epsilon - \delta$ proof of the limit. There's no "leap of faith" here.

And more generally, is there a field of mathematics that treats this kind of "almost equal but not quite" trickiness?

Calculus, using limits.

Where is the study of the almost equal? Please someone ease my cramped eyebrows. I know that calculus makes use of this kind of thing at every possible opportunity, but is there a field of study that distinctly and separately treats those concepts? The study of the infinitesimal yet tangible.

That's an oxymoron, I believe. If something is infinitesimal, it's really tiny, so wouldn't be tangible.

 

[xwolfhunter]

Except at x = 1, the graphs of y = x - 1 and $y = \frac{(x - 1)(x - 2)}{x - 2}$ are exactly the same. The only difference between the two graphs is that the graph of the latter equation has a removable discontinuity -- otherwise known as a "hole" -- at (2, 1).

Yup, got that. And you meant $x=2$.

The expression $\frac{x - 2}{x -2}$ is undefined for only one value: x = 2. As is usually the case where functions become undefined, we see if the limit exist as x gets arbitrarily close to 2. In this case, since $\frac{x - 2}{x -2}$ remains close to 1 for any choice of x other than 2, it seems fairly obvious that the limit should be 1. That assertion can be verified rigorously using the $\epsilon - \delta$ proof of the limit. There's no "leap of faith" here.

Aha, the thing I was looking for: the $\epsilon - \delta$ proof. Thank you.

That's an oxymoron, I believe. If something is infinitesimal, it's really tiny, so wouldn't be tangible.

Isn't the whole point of a limit that it ignores an infinitesimal point, because it can't be made to go away, but it still has tangible effects on what's going on? I'm just guessing, because I haven't read it, but probably the $\epsilon - \delta$ proof will tell me very clearly that in the context of limits, it's okay to ignore an infinitesimally small point. As axmls said, we don't care what happens at a point, which implies that something happens at a point, which implies that something going on at a point can be tangible.

I mean . . . isn't it a very fundamental part of mathematics as a whole that infinitesimals still have properties? Even if we can ignore them.

To paraphrase Horton Hears a Who: "A property's still a property, no matter how small."

 

[Mark44]

Isn't the whole point of a limit that it ignores an infinitesimal point, because it can't be made to go away, but it still has tangible effects on what's going on?

To restate what axmls said,

when taking a limit, we don't care what actually happens at the point (only what happens very close to it).

What you said above, with "infinitesimal point," doesn't make sense. We don't talk about points as being infinitesimal.

Here's an example:
$$f(x) = \begin{cases} x, & x \neq 2 \\ 3, & x = 2 \end{cases}$$

Sketching a graph of this function, it's fairly easy to see that $\lim_{x \to 2}f(x) = 2$, despite the fact that f(2) is defined to be 3. So it is completely immaterial that the point (2, 3) has any effect, tangible or otherwise, on the limit of this function as x approaches 2.


I edited this quote from you...

I'm just guessing, because I haven't read it, but probably the ϵ−δ proof will tell me very clearly that in the context of limits, it's okay to ignore a n infinitesimally small point. As axmls said, we don't care what happens at a point, which implies that something happens at a point, which implies that something going on at a point can be tangible.

"tangible" means perceptible by touch. A point doesn't fit this description.
We don't care what happens at only a single point. I could change the definition of the function to any number whatsoever, and it would have zero effect on the value of the limit. What we care about is what's happening in the intervals around the number in question (2 in this case).

I mean . . . isn't it a very fundamental part of mathematics as a whole that infinitesimals still have properties?

 

[xwolfhunter]

To restate what axmls said,

What you said above, with "infinitesimal point," doesn't make sense. We don't talk about points as being infinitesimal.

Here's an example:
$$f(x) = \begin{cases} x, & x \neq 2 \\ 3, & x = 2 \end{cases}$$

Sketching a graph of this function, it's fairly easy to see that $\lim_{x \to 2}f(x) = 2$, despite the fact that f(2) is defined to be 3. So it is completely immaterial that the point (2, 3) has any effect, tangible or otherwise, on the limit of this function as x approaches 2.


I edited this quote from you...
"tangible" means perceptible by touch. A point doesn't fit this description.
We don't care what happens at only a single point. I could change the definition of the function to any number whatsoever, and it would have zero effect on the value of the limit. What we care about is what's happening in the intervals around the number in question (2 in this case).

What happens at a single point is not tangible (roll with it) because we ignore what happens at a single point . . . but a single point still has properties. That's all. Nothing more.

 

[Mark44]

We don't care what happens at a single point because we ignore what happens at a single point . . . but a single point still has properties. That's all.

You could rephrase what you said. "We ignore what happens at a single point because we don't care what happens at a single point..."

What properties of this single point are you talking about? A point is a pretty simple entity. Geometrically, its dimension (or length) is zero. About the only property we care about is its position, which in the plane is given by two coordinates.

 

[xwolfhunter]

You could rephrase what you said. "We ignore what happens at a single point because we don't care what happens at a single point..."

True.

What properties of this single point are you talking about? A point is a pretty simple entity. Geometrically, its dimension (or length) is zero. About the only property we care about is its position, which in the plane is given by two coordinates.

Eh. Dunno. No longer interested in this discussion. It won't ever go anywhere, and I'm no mathematician yet, so I'm hardly qualified to argue math with you. Pedanticism, I'm good at, but bored with. Thank you for your responses, they had good info and I learned from them.

 

[HallsofIvy]

When you say "we ignore what happens at a single point" you have to specify that you are talking the point at which the limit is being taken. Furthermore, it is not "pedanticism" when it is a detail crucial to the concept.

The definition of "limit" is "$\lim_{x\to a} f(x)= L$ if and only if, given $\epsilon> 0$ then then exist $\delta> 0$ such that if $0< |x- a|< \delta$ then $|f(x)- L|< \epsilon$.

The reason I quote that in detail is to point out the "<" in $0< |x- a|$. What happens when |x- a|= 0, when x= a, is completely irrelevant to the limit at x= a.

 

[xwolfhunter]

When you say "we ignore what happens at a single point" you have to specify that you are talking the point at which the limit is being taken. Furthermore, it is not "pedanticism" when it is a detail crucial to the concept.

The definition of "limit" is "$\lim_{x\to a} f(x)= L$ if and only if, given $\epsilon> 0$ then then exist $\delta> 0$ such that if $0< |x- a|< \delta$ then $|f(x)- L|< \epsilon$.

The reason I quote that in detail is to point out the "<" in $0< |x- a|$. What happens when |x- a|= 0, when x= a, is completely irrelevant to the limit at x= a.

. . . nobody was questioning that. As I said, I'm not a mathematician yet, but I obviously understand that. I don't really get the point of this post.

Edit: Also, you wrote that limit definition really rather more abstrusely than need be . . . what even is that?
Edit edit: Your wording is just whack man. Is english your second language?
Edit edit edit: Never mind, you just stated it differently from how it is in my calc book with a few typos thrown in.

 

[xwolfhunter]

To be honest, when I posted this originally, I was mostly interested in the fundamental difference between $\frac{1}{1}$ and things like $\frac{x-2}{x-2}$. When a variable is constrained by a constant, what is that really? There is a field of possibilities introduced by $\frac{x}{x}$, and suddenly, when the constant is added, there's a little hole in that field. What is the relationship between variables and constants, what general things can be said about their scope when they intermingle, what's the difference between $x-2$ as a numerator and $x-2$ as a denominator? This is what I would like to know more about, not some basic limit definition that I already read in my calculus textbook. (Or, incidentally, anything else that I can read in my calculus textbook, because that is what my calculus textbook is for.)

 

[Mark44]

When you say "we ignore what happens at a single point" you have to specify that you are talking the point at which the limit is being taken. Furthermore, it is not "pedanticism" when it is a detail crucial to the concept.

The definition of "limit" is "$\lim_{x\to a} f(x)= L$ if and only if, given $\epsilon> 0$ then then exist $\delta> 0$ such that if $0< |x- a|< \delta$ then $|f(x)- L|< \epsilon$.

The reason I quote that in detail is to point out the "<" in $0< |x- a|$. What happens when |x- a|= 0, when x= a, is completely irrelevant to the limit at x= a.

. . . nobody was questioning that. As I said, I'm not a mathematician yet, but I obviously understand that. I don't really get the point of this post.

Edit: Also, you wrote that limit definition really rather more abstrusely than need be . . . what even is that?

This is how the term "limit" is defined.

Edit edit: Your wording is just whack man. Is english your second language?

I'm reasonably sure that English is not HallsOfIvy's second language. How is his wording "whack"? What part of it don't you understand?

Edit edit edit: Never mind, you just stated it differently from how it is in my calc book with a few typos thrown in.

What typos? I looked pretty carefully at what HallsOfIvy wrote, and didn't spot any typos. If the definition in your book is appreciably different from what HoI wrote, I'd like to see how it's defined there.

 

[Mark44]

To be honest, when I posted this originally, I was mostly interested in the fundamental difference between $\frac{1}{1}$ and things like $\frac{x-2}{x-2}$.

As I said in post #4, the only difference between 1 and $\frac{x - 2}{x - 2}$ is that the latter is undefined at x = 2. If you graph y = 1 and $y = \frac{x - 2}{x - 2}$, the graphs are identical, except at x = 2. On the latter graph, there is a "hole" at (2, 1).

When a variable is constrained by a constant, what is that really?

What does this mean?

There is a field of possibilities introduced by $\frac{x}{x}$, and suddenly, when the constant is added, there's a little hole in that field.

No, there's a hole in the line -- i.e., a point in the line that isn't there. The term "field" has a mathematical definition. If you're interested, you can look it up on wikipedia.

What is the relationship between variables and constants, what general things can be said about their scope when they intermingle, what's the difference between $x-2$ as a numerator and $x-2$ as a denominator?

In order, variables can take on different values, while the value of a constant doesn't change. The "scope when they intermingle" is covered in basic algebra, for the most part. For example, the expression x + 2 has values that depend solely on the value of x. There's not much more to say about this.

As far as $x-2$ as a numerator and $x-2$ as a denominator, the first one is on the top and the second one is on the bottom. The only point to take away is that you can't divide by zero, which is why $\frac{x-2}{x-2}$ is undefined when x = 2.

The important thing about this fraction is its behavior when x is close to 2, and we use limits to determine this.

Here are a couple of other examples where x - 2 appears in the denominator, both with different behaviors from the rational expression above.
1. $\lim_{x \to 2}\frac 1 {x - 2}$ is undefined
2. $\lim_{x \to 2}\frac 1 {(x - 2)^2}$ is $\infty$. All this means is that $\frac 1 {(x - 2)^2}$ can be made arbitrarily large as x gets closer to 2. It does NOT mean that if we plug in 2 for x, we get $\infty$.

This is what I would like to know more about, not some basic limit definition that I already read in my calculus textbook. (Or, incidentally, anything else that I can read in my calculus textbook, because that is what my calculus textbook is for.)

 

[xwolfhunter]

This is how the term "limit" is defined.
I'm reasonably sure that English is not HallsOfIvy's second language. How is his wording "whack"? What part of it don't you understand?

What typos? I looked pretty carefully at what HallsOfIvy wrote, and didn't spot any typos. If the definition in your book is appreciably different from what HoI wrote, I'd like to see how it's defined there.

Firstly, and only because it's satisfying, please examine more closely:

given ϵ>0 then then exist δ>0

I'm sure I don't need to say much more. Ah hell, I'll say it anyway. "Then then exist"? How did you miss that?

Heh.

As for how it differs from the one I first read, it doesn't. It was just presented in a different order, and was much less condensed (it's a calculus textbook for college students), so it took a few read-throughs for me to reconcile the two.

As I said in post #4, the only difference between 1 and $\frac{x - 2}{x - 2}$ is that the latter is undefined at x = 2. If you graph y = 1 and $y = \frac{x - 2}{x - 2}$, the graphs are identical, except at x = 2. On the latter graph, there is a "hole" at (2, 1).
What does this mean?
No, there's a hole in the line -- i.e., a point in the line that isn't there. The term "field" has a mathematical definition. If you're interested, you can look it up on wikipedia.
In order, variables can take on different values, while the value of a constant doesn't change. The "scope when they intermingle" is covered in basic algebra, for the most part. For example, the expression x + 2 has values that depend solely on the value of x. There's not much more to say about this.

Awesome, I was conceptually asleep throughout high school math, so I'll definitely pick up a good algebra book. Any you would recommend that would not be offered in a high school class? Like, designed for people who are interested in concepts not calculations?

As far as $x-2$ as a numerator and $x-2$ as a denominator, the first one is on the top and the second one is on the bottom. The only point to take away is that you can't divide by zero, which is why $\frac{x-2}{x-2}$ is undefined when x = 2.

Helpful.

Edit: If this is going to go on much longer, please be aware of a few things. I did get through high school math. I learned precalculus in a couple of weeks, got a 4 on the AP calc exam, I memorized 400 digits of pi, blah blah. I am discussing things right now that I nominally learned years ago, but I never conceptualized them because hey, I'm smart enough to have coasted by with no effort and ended up cum laude, and that's why I'm here - reading through my calc book I'm finding all kinds of things I should already have conceptualized but haven't. If a good algebra book can get me past it, I'm all brains, and soon I'll be talking topology and combinatorics with you. I'll probably be less irritating then. Maybe help me get there?

 

[Mark44]

given ϵ>0 then then exist δ>0

"Then then exist"? How did you miss that?

I guess my eyes skipped right over the second "then" and read it as HoI intended, "there exists".

Awesome, I was conceptually asleep throughout high school math, so I'll definitely pick up a good algebra book. Any you would recommend that would not be offered in a high school class? Like, designed for people who are interested in concepts not calculations?

I don't think you'll find any, unless perhaps the title is "Algebra for Poets."

 

[xwolfhunter]

I guess my eyes skipped right over the second "then" and read it as HoI intended, "there exists".

Right. My eyes don't really do that too often.

I don't think you'll find any, unless perhaps the title is "Algebra for Poets."

Heh. Got a chuckle from me.

Alright well I'll either learn from osmosis or find some allegorical masterpiece. Thanks for your responses, they were partly helpful. Earlier.

 

[micromass]

Awesome, I was conceptually asleep throughout high school math, so I'll definitely pick up a good algebra book. Any you would recommend that would not be offered in a high school class? Like, designed for people who are interested in concepts not calculations?

A very good algebra book that focuses on the concepts is Gelfand and Shen's "Algebra". If you know some mathematics, then you'll recognize the name Gelfand from miles away. Yes, a top mathematician like Gelfand actually wrote a high school algebra text! As a result, the text is brilliant. While the material developed is mostly easy, the conceptual way Gelfand introduces it is superb. (If you like it, Gelfand has other high school books).

 

[xwolfhunter]

A very good algebra book that focuses on the concepts is Gelfand and Shen's "Algebra". If you know some mathematics, then you'll recognize the name Gelfand from miles away. Yes, a top mathematician like Gelfand actually wrote a high school algebra text! As a result, the text is brilliant. While the material developed is mostly easy, the conceptual way Gelfand introduces it is superb. (If you like it, Gelfand has other high school books).

Actually, I only recognize Gelfand from the name Boris Gelfand, a current-ish top-level chess grandmaster.

I will look into that for sure, thank you very much!