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Understanding "Equality" notation in a limit

Talks about a limit is not a value that can be achieved, but it's a value that can be arbitraly approached to.

So my question is then, the formular pi*R^2 for the area of the circle is not an exact value. But only a limiting value(?).

Just like the 2 is never achieved in the following.

$$\lim_{k\to \infty} \sum_{n=0}^k \frac{1}{2^n} = 2$$

So we can say, we only know the limiting value for the area of the circle, we don't know the exact value of the area of a circle? (Because the only way I know to measure the area of a circle is by summing many small rectangles)

  • edit - in response to VTMcan 's answer

I thought you can never reach 2 for the sum. (How can you prove it?) How about the following.. I don't think you can say limiting value is the exact value when a limit exists.

$$\lim_{k\to \infty} \frac{1}{2^n} = 0$$

eugene
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  • How do you define the area of a circle, though? You will find that you need to accept limits in the definition, one way or another. What does $A$ metres squared even mean? There isn’t any way to draw $A$ square metres inside many shapes with area $A$, yet we still declare that to be their area. – FShrike Aug 28 '22 at 08:26
  • @FShrike your comment seems to imply that the answer to my question is yes, it's only a limit. Because the definition of the circle area involves limit.? – eugene Aug 28 '22 at 08:34
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    Not, “only”, a limit. What’s wrong with limits? You need an understanding of them to define a large chunk of interesting and useful mathematics. They’re as “real” as “exact” equations! A limit is just a way to describe a number which already exists. How do you define the area of a circle? It’s not an easy question – FShrike Aug 28 '22 at 08:39
  • Nothing wrong with limit. I should probably have to say, it's a limit. not an exact value. I can see they are 'exact' equations. but I can't still see they are exact value. I can see they are a way of describe a number. And it seems, my question can be rephrased as 'is there another way? more concrete way to define the number?' and the answer seems no. – eugene Aug 28 '22 at 08:53

2 Answers2

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(One more try...) You're continually saying that you don't see things that nobody ever said you should see.

Fact: $\lim_{n\to\infty}\frac1n=0$.

That limit is exactly zero. Now you object

Objection: "But I don't see it - we never actually get to $0$", or words somewhat to that effect.

The answer to the objection is that it's simply no objection at all; the fact that $\frac1n\ne0$ has precisely as much relevance here as the fact that Botswana is not located in Ohio. Because the Fact has a precise definition, and whether or not $\frac1n=0$ simply does not come up:

Definition: The Fact means that for every $\epsilon>0$ there exists $N$ such that $\left|\frac1n-0\right|<\epsilon$ for every $n>N$.

Of course dealing that definition can be confusing. But for now just note that you don't see anything in the definition that talks about whether $\frac1n=0$.

David C. Ullrich
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  • Thanks for taking time. So then 1 / lim (1/n) is undefined? Since lim (1/n) is exactly 0. I thought 1 / lim (1/n) is infinity.. I understand if you don't answer; – eugene Sep 01 '22 at 08:32
  • Yes, $1/\lim(1/n)$ is undefined. Because it's exactly the same thing as $1/0$, and there's no such thing. – David C. Ullrich Sep 01 '22 at 12:42
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Limit values are exact values if the limit exists and is well-defined.

That sum does not arbitrarily approach 2, it is equal to 2. Similarly, the area of a circle is exact and finite.

VTMcan
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  • Hi, I've edited the original question, and added my reasoning (a counter example) about your "Limit values are exact values if the limit exists and is well-defined." – eugene Aug 28 '22 at 08:33
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    You seem to still be confused based on your edits. Limits are exact values when well-defined. "Infinitely approaching" IS equal to, I do not think you are grasping infinity. Infinity does not work as some discrete function that slowly approaches a value, all of infinity automatically calculates at once and that infinite sum/calculation is equal to an exact value. – VTMcan Aug 28 '22 at 08:34
  • Yes that limit is still exactly equal to 0. That is not in any way a counter example. – VTMcan Aug 28 '22 at 08:35