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I'm studying my first year of Mathematical Engineering and I had Calculus I. I was hopping to get some answers to some questions I had when learning Calc..

I was taught pre-Uni. that a limit is just the concept of getting as close to a number $x$ as you can get, without actually getting to said value. Say we have to evaluate the limit:

$\lim_{x\to4} f(x)=x^2\$ $_{(1)}$

So I would interpret this as: As $x$ approaches $4$, in the function $f(x)=x^2$ then the result will approach $16$, therefore the solution for this limit is $16$. So we actually never get to $16$, we just get as close as we can be. The same way that if we have:

$\lim_{x\to4} f(x)=\frac{x-4}{x-4} = 1$ $_{(2)}$, when $x = 4$ in $f(x)=\frac{x-4}{x-4} = \frac{0}{0}\neq 1$ = Undefined$ $_{(3)}$

After Limits, they thought me about continuity and I started to see the usefulness of limits, and how they are used when defining the continuity of a function, and what happens we evaluate them from the right or left, etc... . All that good stuff.

I come to Uni. and I have a specific class for Calc. and here they gave me a more concrete definition for a limit. Here it is:

For all $\epsilon > 0$ it exists a $\delta > 0$ so that for all $x$, if $x$ $0<|x-a|<\delta$ then $|f(x)-L|<\epsilon$ $_{(4)}$

(If in English we use a different way of conveying the same definition please let me know so I can change it)

Now, correct me if I'm wrong but what this definition is saying is that if we first take an arbitrary point $a$ and $\lim_{x\to a}f(x) = L$ in the function $f(x)$, then if we create an "environment" (delta) around $a$, we can always say that the value of $L$ will be inside another "environment" (epsilon) around L in the $y$ coordinates.$^1$

Alright, a more mathematical rigours definition of a limit, which was followed up with a more rigorous definition of continuity, and how to define it and etc... .Everything seems to work to me, until we moved to derivatives.

Now, the issue I have is with the definition of a derivative, and not I know how to do derivatives and what they represent and so on, however my question is bout the definition of it.

$\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$ $_{(5)}$

Now, the issue is that the definition is in terms of a limit, but a moment before I just explain that as far as my knowledge goes, a limit doesn't actually reach the value, otherwise you would get weird cases like $(3)$. Furthermore, that's why we can get rid of the divisor $h$ in further simplification when working with the definition, as it's not $0$, just the limit of h approaching $0$.

This question persisted when I was taught about integrals, specially when learning about the Riemann sum. If the rectangles get thiner and thiner, but still are rectangles (aka the side the shrinks never gets to 0) they still aren't the exact area under said curve. If the rectangles would eventually transform into lines, I would understand how that could represent all the space under a curve.

Now, I've asked my professors about this and had the opportunity to have a long conversation about this with them. Here are the responses in case it helps:

Professor 1: After talking with my calculus professor, she told me that this question needed someone with more mathematical knowledge (she is a physicist).

Professor 2: I had a lengthy discussion with my Linear Algebra professor, and came to the following conclusion. A limit, is indeed an approximation, and the derivative or area under a curve are so too. However, the definition of an Area, or the reate of change of function can be defined as both a value, say $s$, and/or the "Aspiration" of said area/rate of change. Say we take a look at the following definite integral:

$\int_{1}^{3}2x dx = 8$ $_{(6)}$

So 8 would be the value that infinite series of sums approaches, therefore it is its "aspiration", and because the definition of an area can also be define as the "aspiration", the area under the curve is 8. Now, he also told me that there are cases in which the area is just a value, say $s$, say, for example:

$\int_{1}^{3}1 dx = 2$ $_{(7)}$

No "aspiration" whatsoever, just good old 2.

My Take: Alright, so what I believe is that, both the derivative and integral aren't approximation of any kind, no "aspiration", no other kind of saying approximation. They do represent the actual area under the curve, or the change of rate of a function. And the limit is indeed an approximation, the thing is that when you infinitely approximate something, you actually get to that something. Now, if that's the case, I want to know how we came to that conclusion.

Foot Notes:

  • $^1$ I've said this to my teacher and she seemed to agree with that statement, however something about it doesn't feel right. So in case I'm utterly wrong, please try to see the bigger picture of what I'm trying to say. What this question is really about is that if we infinitely approach something, do we get to that something? If so, always? Why? How do we know that?. If not, I believe there are cases that in which we are using it as it was. Are they right? Why? How do we know that?

Important Notes:

  • I study in Spanish, this means that I had to make a rough translation? in order to post this post. Some technicalities might be wrong, and I might have used a different name to refer to the same mathematical concept (let alone the grammatical errors I might have written). If you think I did please let me know and I'll happily change it.
  • I don't know why, but when I write a limit in LaTeX form, the info. just won't go under it. I don't know if this way of writing it has a different meaning so please, just assume every time to see it like that, know that it was a typo.
  • If someone has read this whole post, I'm very VERY thankful. It isn't specially short, and it could have been even longer so thank you once again.
amWhy
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MMarqueto
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    I think you have things backwards. If, say, $\lim_{x\to a}F(x)=L$ then $L$ is a well defined value. Taking $x_0$ near $a$ and then looking at $F(x_0)$ is an approximation. For a geometric example, the circumference of the unit circle has length exactly $2\pi$. That can be viewed as the limit of the perimeters of regular $n-gons$ inscribed on the circle for large $n$....in that case, such a perimeter is an approximation to the length, the limit is the exact value. – lulu Dec 30 '23 at 21:20
  • About your words just below $(5)$, "a limit doesn't actually reach the value, otherwise you would get weird cases like $(3)$." I think the meaning of "the value" here is ambiguous: for line $(3)$, the limit $\lim_{x\to4}\frac{x-4}{x-4}$ does reach and is exactly $1$, even though $x$ does not have to "reach $4$". – peterwhy Dec 30 '23 at 22:21
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    I haven't read much past the first paragraph, but the title seems to be a misconception. A limit is not an approximation. – Xander Henderson Dec 30 '23 at 22:58
  • In a sense, mathematically there isn't a prior a precise meaning of the area under a curve apart from saying it intuitively should be approximately by the sum of areas of rectangles, whose areas we can all agree are base times height. So, we just end up defining that limit to be the area under the curve. You might also see the concept of Lebesgue measure in the future, which provides another way of defining this area, but in the end it's really just a fancier way of approximating that area using rectangles. – David Gao Dec 30 '23 at 23:10
  • So in a sense, all areas (apart from those of triangles) are defined to be limits. So, in this case the limit does reach that end result, because that is how it is defined to be. (Beware though, that there are cases in which what may intuitively look like an infinite approximation of something ends up not agreeing with the formal definition, even when the latter is also a limit. For an example, see this question: https://math.stackexchange.com/questions/12906/the-staircase-paradox-or-why-pi-ne4) – David Gao Dec 30 '23 at 23:15
  • Limit of function $f$ on $x_0$ is usually defined so as not to bother about the value of $f$ on $x_0$. This allows cases where $f$ is not defined on $x_0$, as you pointed out. That's convenient: there are cases where it allows "extending $f$ by continuity" on $x_0$. Also that makes it more similar to the case where limit is taken on $+\infty$ or $-\infty$. However that's purely a convenient convention, and the opposite convention, i.e. taking into account the value of $f$ on $x_0$, is sometimes adopted; cf. https://en.wikipedia.org/wiki/Limit_of_a_function#Deleted_versus_non-deleted_limits. – Jean-Armand Moroni Dec 30 '23 at 23:17
  • What is $\lim_{x \to 0} \frac{\sin(x)}{x}$? The answer is $1$. As $x$ approaches $0$, $\frac{\sin(x)}{x}$ approaches $1$. So the limit is $1$. – littleO Dec 31 '23 at 06:00
  • Since you asked about the translation of terms, what you called an "environment" is called in English a "neighborhood". (Your translation appears to be very accurate and understandable nevertheless.) – David K Dec 31 '23 at 07:44
  • Pratically, you cannot avoid to do approximation, whatever you do. Limit is a result good for every approximation (for all \epsilon...etc). – lib Dec 31 '23 at 08:23
  • See related https://math.stackexchange.com/a/1954854/72031 – Paramanand Singh Jan 03 '24 at 00:40
  • A much easier question that you could try to focus on is: why $0.9999...=1$? See wikipedia's page https://en.wikipedia.org/wiki/0.999... . This looks extremely close to what you are asking, so I suggest to think about it – Lorenzo Pompili Jan 03 '24 at 02:24

1 Answers1

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You have seen correctly that a limit is not an approximation: it is the end result itself. The limit is what is “reached”, not some step along the way. The approximations are what you get during the process of passing to the limit, but the actual limit is meaningful on its own and is approximating nothing.

Example. Consider $\pi$. It is a specific number, not an approximation of anything. When we write $\pi = 3.1415926…$, the truncated decimals $3$, $3.1$, $3.14$, $3.141$, and so on are approximations of $\pi$, but the full decimal expansion of $\pi$ is exactly $\pi$, not an approximation of something.

In Hairer and Wanner’s great book “Analysis by its History”, they summarize the realization in the 19th century about what all the objects in calculus are:

What is a derivative? A limit.

What is a definite integral? A limit.

What is a limit? A number.

Then we are led to the hardest question:

What is a number?

This was answered by rigorous constructions of real numbers, which were first done in the late 19th century by Dedekind (using Dedekind cuts of rational numbers) and Cantor (using equivalence classes of Cauchy sequences of rational numbers).

By the way, a limit is not only a number: you could think about limits of functions, curves, etc. But in the development of analysis, the first limit concept that needs to be understood is limits of numbers.

KCd
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  • Hello! Thank you for answering. The Q. was closed so I ignore if this will reach you. You said something very interesting that I didn't include. Pi cannot be represented by any rational number base, however it can be calculated by the limit of a series of approximations in said base. My original Q. didn't emphasise this enough. What I'm really after is the answer to how do we know that by infinitely approaching something, we get to that something. The limit is not an approx., but the result of infinitely approx. that actually get do the "actual" value. How is it that we know that is true? – MMarqueto Jan 04 '24 at 01:51
  • You need to know the error is going to 0. The numbers 3, 3.1, 3.14, 3.141, ... are getting closer to 4 at each step but certainly don't get to 4 in the limit. – KCd Jan 04 '24 at 02:06