Nature of Limits: Learn Calculus Basics w/o Limits Distraction

[NoahsArk]

The question about how using limits can give us the exact slope of a line tangent to a curve is something so far I haven't quite been able to grasp. I do intuitively understand how using limits can get us so close to the exact slope that any difference shouldn't matter in the real world because we can get as close as we want. The answers I've seen so far to this question about why limits give us an exact slope seem to go beyond calculus and are more related to complex logical proofs. My question now is should I even be getting hung up on it when I'm learning calculus? If a student of calculus has a basic understanding of derivatives and integrals, and knows how to solve standard calculus problems, is it just a distraction to be thinking about things like this? I also don't understand the solution to Xeno's paradox, which seems like it also involves an understanding of limits, but that also seems like a distraction to be thinking about in learning calculus.

I've noticed in my studying of different subjects I often will go down a culdesac in trying to understand exactly how something works, and this comes at the expense of not learning the basic material of the subject. If for, example, I am able to solve basic calculus problems like finding derivatives, finding minimum and maximum points, and if I understand the general idea of what derivates are, does it matter that I don't understand exactly how limits give us the exact slope of a tangent line (or how they help give us the exact area under a curve as opposed to a super close approximation)? Calculus problems in school and online lessons do not ask the student about the nature of limits. They do ask students to find what the limit is of a certain function as x approaches a value, and those are easier for me to deal with than to think about how limits work on a deeper level.

 

[Drakkith]

My question now is should I even be getting hung up on it when I'm learning calculus? If a student of calculus has a basic understanding of derivatives and integrals, and knows how to solve standard calculus problems, is it just a distraction to be thinking about things like this?

Yes, it's mostly a distraction if you're simply trying to get through a class and move on with your education. The idea of a limit works, it is useful, and it is relatively simple to explain (if not prove) to students. Get through your classes first and then worry about the underlying proofs.

I also don't understand the solution to Xeno's paradox, which seems like it also involves an understanding of limits, but that also seems like a distraction to be thinking about in learning calculus.

Which version of the paradox and which solution are you referring to?

 

[NoahsArk]

Which version of the paradox and which solution are you referring to?

The one where to get from point A to point B you have to go half way, then half of the rest of the way, etc. and how you still reach the other side. I don't have any particular solution in mind, I just remember not understanding the solutions that I looked up.

 

[Drakkith]

The one where to get from point A to point B you have to go half way, then half of the rest of the way, etc. and still reach the other side.

Okay. What about the solution don't you understand?

 

[Mark44]

I also don't understand the solution to Xeno's paradox, which seems like it also involves an understanding of limits, but that also seems like a distraction to be thinking about in learning calculus.

You're probably not far enough along yet to grasp Zeno's Paradox (note that it's Zeno, not Xeno). To understand why 1/2 + 1/4 + 1/8 + ... adds up to 1, in the limit, you need to have an understanding of infinite series, but this topic usually doesn't come along until the 2nd semester or 3rd quarter of calculus.

 

[PeroK]

The question about how using limits can give us the exact slope of a line tangent to a curve is something so far I haven't quite been able to grasp. I do intuitively understand how using limits can get us so close to the exact slope that any difference shouldn't matter in the real world because we can get as close as we want. The answers I've seen so far to this question about why limits give us an exact slope seem to go beyond calculus and are more related to complex logical proofs. My question now is should I even be getting hung up on it when I'm learning calculus?

Whether it matters or not is a moot point, but the bit I've underlined is very wrong. 1) This is mathematics, so there is no real world; 2) The tangent can't be anything else but the limit. If you take any slope except the one calculated by the limit, then that is definitely not the tangent.

What you are really saying is that, for example: $\frac 1 2 + \frac 1 4 + \frac 1 8 \dots \ne 1$. That's it's not really $1$, that it's just "close to 1" and that is very wrong. It is exactly equal to $1$.

It can't be any else other than $1$. Because if you take any number less than $1$ then that sum is definitely greater than that number. In short:$$\frac 1 2 + \frac 1 4 + \frac 1 8 \dots = \lim_{N \rightarrow \infty} \sum_{n = 1}^{N} \frac 1 {2^n} = 1$$
No ifs, buts or maybes and no "real world" that has an answer different from $1$.

 

[PeroK]

PS let's take a better example: the graph $y = x^2$. The tangent at the origin is definitely the $x-$axis. It can't be either an upward or downward slope. It must be a horizontal line.

We can calculate that by taking the limit of the approximate tangents. Each of these approximate tangents is an upward or downward slope, but the limit (the tangent itself) is horizontal.

Now, imagine any graph with a tangent at a particular point. We may orient the graph so the point is the origin and the tangent is horizontal. And the same applies: the tangent is the limit of approximate tangents. The limit itself is not an approximation.

 

[Delta2]

note that it's Zeno, not Xeno

Not sure about the Z or X but his full name contains an "n" at the end that is "Zenon". In ancient Greece many males names were ending at "-on" (as well as -os or -is e.g Alexandros, Aristotelis, which are used in modern Greece as well).

 

[Delta2]

The question about how using limits can give us the exact slope of a line tangent to a curve is something so far I haven't quite been able to grasp. I do intuitively understand how using limits can get us so close to the exact slope that any difference shouldn't matter in the real world because we can get as close as we want. The answers I've seen so far to this question about why limits give us an exact slope seem to go beyond calculus and are more related to complex logical proofs. My question now is should I even be getting hung up on it when I'm learning calculus? If a student of calculus has a basic understanding of derivatives and integrals, and knows how to solve standard calculus problems, is it just a distraction to be thinking about things like this? I also don't understand the solution to Xeno's paradox, which seems like it also involves an understanding of limits, but that also seems like a distraction to be thinking about in learning calculus.

I've noticed in my studying of different subjects I often will go down a culdesac in trying to understand exactly how something works, and this comes at the expense of not learning the basic material of the subject. If for, example, I am able to solve basic calculus problems like finding derivatives, finding minimum and maximum points, and if I understand the general idea of what derivates are, does it matter that I don't understand exactly how limits give us the exact slope of a tangent line (or how they help give us the exact area under a curve as opposed to a super close approximation)? Calculus problems in school and online lessons do not ask the student about the nature of limits. They do ask students to find what the limit is of a certain function as x approaches a value, and those are easier for me to deal with than to think about how limits work on a deeper level.

Not sure if I understand your problem correctly, however yes you can do derivatives and integrals without worrying about the concept of the limit that lies behind those two basic concepts of calculus. Derivative is the limit of the rate of change, while integral is the limit of a sum of infinitely many, infinitely small quantities. I sense that there is something you don't understand correctly about the concept of the limit but can't tell exactly what it is. Intuitively the limit is that we can get our function to be as close as we want to a certain value, as long as the independent variable of the function gets close enough to a certain point. BUT yes as i said at start you don't have to worry a lot about the concept limit, you can do calculus of derivatives and integrals without it.

 

[NoahsArk]

I had to learn a bit of latex to write this reply:)

What you are really saying is that, for example: $\frac {1}{2} + \frac {1}{4} + \frac {1}{8}...$⋯≠1. That it's not really 1, that it's just "close to 1" and that is very wrong. It is exactly equal to 1.

It can't be any else other than 1. Because if you take any number less than 1 then that sum is definitely greater than that number. In short:

$$ \frac {1}{2} + \frac {1}{4} + \frac {1}{8}... = \lim_{N \rightarrow +\infty} \sum_{n=1}^\infty \frac {1}{2^n} = 1 $$

No ifs, buts or maybes and no "real world" that has an answer different from 1.

To clarify, I am not saying that it doesn't exactly = 1. I am saying that at this stage, I don't understand why it exactly equals 1, and only understand that before it reaches 1 it's getting very very close to 1. My question was: should I for now just accept the fact that it equals 1 without worrying about understanding how to prove it? My sense from reading the responses is that I can just accept it as a fact for now.

The way you stated the problem though is a helpful way for me to think about it. Is it the case that $\frac {1}{2} + \frac {1}{4} + \frac {1}{8}... =$ exactly 1 for the same reason that limits give the exact slope of a line tangent to a curve?

What about the solution don't you understand?

I don't remember what the solutions that I read said- I only remember that I didn't understand them. I just brought it up because I want to know if it's even necessary to understand at this stage.

To understand why 1/2 + 1/4 + 1/8 + ... adds up to 1, in the limit, you need to have an understanding of infinite series

I haven't studied infinite series, but it's good to know about the topic.

 

[Mark44]

PS let's take a better example: the graph $y = x^2$. The tangent at the origin is definitely the $y-$axis.

I think you really meant x-axis here.

 

[Gaussian97]

Well, here I think it's important to distinguish between two things. One thing is to know how to compute a limit, while the other is know what a limit actually is. And I said that, which maybe you see kind of obvious, because in my country is very common that people learn in High School how to compute limits, but nobody tells them what a limit is. And only those who study calculus in their degree finally see the correct definition of a limit.
The problem is that a lot of students in High School try to understand the limit, not only learn how to compute it, and they end up in exactly the same situation as you. So my first question to you is: Have you been introduced what a limit is? Or you only have been told how to compute them?
If you haven't seen the formal definition of a limit, don't worry if you don't understand it, it's very difficult. You can keep computing limits and hoping somebody will explain to you what they are in the future, or look at the formal definition and try to understand it.

If you are taken a course in calculus and you have seen the formal definition of a limit, then that's why a limit is what it is. Is as simple as that. Why the limit is 1? Because when you put it in the definition, you get a True statement. If you put any other number in the definition you get a statement that is False. There's no deeper reason.

Of course, the discussion could continue on why the definition is what it is and not another, or things like that. But as soon as you accept the definition of a limit, any time you ask why the limit is 1, the only answer is: by definition.

 

[NoahsArk]

So my first question to you is: Have you been introduced what a limit is? Or you only have been told how to compute them?

I'm not taking a calculus course in school. I'm just learning on my own partly to gain knowledge, and partly to keep more career options open. I also have a daugher who will be in high school next year, so knowing calculus will allow me to help her more.

I have learned more or less how to compute limits (e.g. direct substitution, factoring, etc.). I have also read up on what a limit is, but that's the part I don't fully understand.

Of course, the discussion could continue on why the definition is what it is and not another, or things like that. But as soon as you accept the definition of a limit, any time you ask why the limit is 1, the only answer is: by definition.

I was reviewing the definition this week, the "epsilon delta definition". It actually made some sense to me. My main problem has been why the limit gives exact answers to calculus problems. For example, in many limit problems, the original equation has an x in the denominator, and you have to try and change the equation into a new form to get x out of the denominator so that you can plug a certain value into x and get the limit. Basically you end up with two forms of the same equation, with the same graph, except that the first equation has a gap in it where x at a certain value is undefined because it would've caused a zero to be in the denominator. When you have the new equation it becomes clear why we are getting an exact value. but what's not clear is why we are allowed to change the equation from it's original form.

 

[PeroK]

I have learned more or less how to compute limits (e.g. direct substitution, factoring, etc.). I have also read up on what a limit is, but that's the part I don't fully understand.

Here's one way to look at it. Suppose we start with the sequence: $$1, \frac 1 2, \frac 1 3, \frac 1 4 \dots$$
We know that the natural numbers never terminate, so we can have an infinite number of terms in that sequence. Note that we can never write them all down, but there is no point at which we are forced to stop. And that's what an infinite sequence is. It goes on forever and we never reach a final term.

If you look at that sequence and ask yourself: is the number $0$ relevant to that sequence in any way?

Now, you may answer "no". You may say that none of the terms in that sequence is zero, so zero has no relevance to that sequence. You might say that since we never reach zero, it has nothing to do with that sequence.

But, you shouldn't be satisfied with that answer. You ought to think that zero is a unique number of special significance to that sequence; that in some sense zero is the limit of that sequence.

At this point you could simply declare - without a mathematical definition - that zero must be the limit of that sequence, in the sense that the sequence gets closer an closer to zero. And you might be able to do the same for many simple sequences: have an intuitive idea of the concept of a limit. Note that a limit is a single number associated with the sequence; it is not part of the sequence and it is not a process. It is a real number.

The hard bit is formulating a precise mathematical definition of a limit. That has only been around for a couple of hundred years. You may be able to get away without understanding that at this stage, but you need the intuitive concept of a limit, I think.