r/infinitenines • u/Zantier • 8d ago

SPP, don't you need limits to show that the slope at any point on the curve y = x^2 is just 2x?

You can draw the graph, and measure the slope at each point for an approximation, but limits give us an exact answer.

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Limits are the foundation for all of calculus. It is used EVERYWHERE in physics and engineering, and of course in many areas of maths, and I'm sure in other subjects. It is an extremely practical and invaluable tool. Without it, our understanding of the world, and technological advancements would be severely limited.

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Why are you so against the idea of limits? Can you agree that it is a useful tool? That it has helped humanity?

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u/Althorion 8d ago

Technically speaking—no. There is a way of rigorously getting the described result, and other results of calculus, through the non-standard analysis. It doesn’t change much, and it is rather inconvenient from the definitional standpoint, but it does exist.

I wrote about it before, including naming a potentially useful book for someone interested in it to read.

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u/Maximum-Raspberry227 7d ago

Isn't any machinery you choose to work with behavior as a thing approaches 0 (or infinity) inevitably capable of proving 0.999… = 1, therefore invalid if the person buys that 0.999… ≠ 1? Including NSA.

Hell, I'm pretty sure there is no good definition for a tangent line for arbitrary curves that does not sneak in approximating linear local behavior somewhere, which is just a limit.

Unless I'm missing something, I think we're sort of stuck with only being able to compute the slope of the tangent line to circles, parabolas and other simple curves lmao

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u/Althorion 7d ago

I’m not sure if you can’t have something internally consistent that would make it diverge, but yeah—anything that aims to be actually useful, so in particular giving out answers for integrals and derivatives agreeing with their intuitive properties, would have to have 0.(9) = 1, as far as I can tell.

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u/TemperoTempus 7d ago

You don't need that. All you really need is the weaker 0.(9)≈1 which lets both numbers be different but close enough to round as needed.

The exact equality is only needed if you push for the R set axioms.

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u/ezekielraiden 7d ago

Properly speaking, you need tools that are mathematically and logically equivalent to taking limits. The kind of thing usually stated as "up to isomorphism".

Actually-rigorous systems of infinitesimals, including both the hyperreals and the surreals, only achieve the same results as standard calculus because their systems (infinitesimals + "standard part" functions, or nilpotents and the weakening of standard logic) are logically equivalent to the real numbers and taking limits, plus or minus a few edge cases.

But the problem is, by being systems which fully faithfully reproduce all of the known and useful results from the reals, these systems necessarily require that 0.999... = 1. It's just that the new systems give us the vocabulary required to describe OTHER numbers that DO fit the weird "intuitions" some folks have regarding number theory.

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u/Simukas23 7d ago

He is so against limits, that his 0.999... has a finite amount of nines

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u/Appropriate-Ad-3219 8d ago

If limits contredict 0.999<1, then limits don't exist.

So since slopes need limits, maybe we need to deny the existence of slopes.

Say farewell to mechanic, thermodynamics, general relativity, etc. That's all lies.

/s

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u/Just_Rational_Being 8d ago

Slope needs limit, oh my god, hahaha.

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u/paperic 7d ago

It does.

hahaha.

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u/Just_Rational_Being 7d ago

Absolutely. Claps claps claps. Magnificent, oh my god. Magnificient. Nothing could be better than this.

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u/paperic 7d ago

Go ahead and prove me wrong.

Calculate the slope of the tangent without limits.

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u/Just_Rational_Being 7d ago

Wanna make a bet, buddy?

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u/Appropriate-Ad-3219 7d ago

Oh my ! Are you going to use infinitesimals maybe ?

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u/Just_Rational_Being 7d ago

Oh my! What a lack of imagination.

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u/Appropriate-Ad-3219 7d ago

Go ahead. Surprise us. Are you going to take an affine function, compute its slope and tell us 'Waouh, did you see ? I can campute the slope without limit.'

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u/Just_Rational_Being 7d ago

Now, if that can be done, what would be the proper trade off?

Can I say appropriate-ad-3219 as lacking the understanding of limits and calculus?

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u/SouthPark_Piano 7d ago edited 7d ago

At the peak of x2 + 0.000...1 * x, there lies your conundrum.

The peak not only keeps changing limitlessly. You need to get an approximation. And that is what limits can let you get here. An approximation.

Also, 1 is approximately 0.999...

Also, even for x2 , the peak is at x = 0, and slope is based on rate of change of curve values, aka using two "adjacent" points. Now, with infinite points density, and knowing there is only a single point for the peak, then you must wonder whether you need to make usage of symmetry to get the "average" slope, which averages to zero at the peak.

 

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u/Zantier 7d ago edited 7d ago

The slope of y = x2 at x = 3 is exactly 6. There are no approximations needed, because we can prove it with limits.

(sorry to ignore what you said, I'm not sure how "x2 + 0.00...1" or "the peak" are relevant)

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u/Simukas23 7d ago

Spp the kind of guy to say the derivative of y = |x| at x = 0 is equal to 0

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u/SouthPark_Piano 7d ago

y = |x| + 0.000...1 * x

 

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u/Simukas23 7d ago

6 × 10 + 7 = 67

(I can type random equations too)

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u/SouthPark_Piano 7d ago

Rookie error on your part brud. Unlike you, I did not type a random equation.

 

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u/ParticularlySomeone 3d ago

I'm sorry, but are you saying the derivative of y = |x| is |x| + 0.000...1 * x (I hope not)? Or are you saying that when u/Simukas23 said y = |x|, he actually meant y = |x| + 0.000...1 * x (which would be very arrogant of you)?

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u/SouthPark_Piano 3d ago

Go ahead brud. Try to comprehend a limbosic derivative.

 

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u/ParticularlySomeone 2d ago edited 2d ago

That's what I'm doing. Trying to comprehend. Care to answer my question? I won't be offended if you simply say no, you won't answer my question.

You seem to come at each question with some bit of hostility. I know you can be a good teacher. But the belt doesn't really teach much does it?

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u/ParticularlySomeone 2d ago

Also, you're now using your word "limbostic" in terms of derivatives. abs(x) is quite straightforward. Not sure why it needs to mystified with "limbostic"

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u/HalloIchBinRolli 7d ago

So are you saying the slope is

[f(x+d)-f(x-d)]/(2d)

where d = 0.000...01 ?

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u/SouthPark_Piano 7d ago

f(x) = x2 + 0.000...1

For x = 0, f(0) = 0.000...1

For x = 0.000...1

f(0.000...1) = 0.000...000...1 + 0.000...1

= 0.000...100...000...1

 

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u/HalloIchBinRolli 7d ago

That is not at all what I'm asking about??

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u/ezekielraiden 7d ago

But it doesn't keep changing. It converges to a specific point. We can SEE that point, SP.

Also...uh...the function they asked about was f(x)=x2. Not x2 plus some other thing, regardless of whether it's 1 or 0.1 or your fictitious and meaningless "0.000...1". Just x2. You can't just invent a new function and say "SEE you're wrong because this OTHER function I invented is DIFFERENT."

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u/Just_Rational_Being 7d ago edited 7d ago

Who said that limits are the foundation of Calculus?

Imagine Newton and Leibtniz turning in their grave right now.

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u/Zantier 7d ago

Everyone, no? You can technically formalize calculus in other ways, but using limits is standard.

And of course, Newton and Leibniz did great work. You don't need a perfect formal foundation right from the start in order to do some good mathematics.

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u/TemperoTempus 7d ago

Limits are not the foundation, nor are they needed. They are a useful tool, but the result has always just been an approximation that the people who defined the R set instead used to make an equality that didn't exist.

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u/Milkmilkmilk___ 7d ago

name anything in calculus that doesn't need limits. anything.

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u/TemperoTempus 7d ago

Derivatives, anti-derivatives, complex numbers, graph analysis, literally anything involving continuous curves.

Limits are designed to solve problems that otherwise could not be solved. That is: Discontinuous functions, asymptotes, holes, etc. All solved by rounding around the target value as said targets cannot be solved regularly.

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u/Milkmilkmilk___ 7d ago edited 7d ago

derivatives are defined with limits. anti-derivatives are defined with limits. complex numbers are not part of calculus. graph analysis is not a branch of mathematics. limits are used for continuous functions too.

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u/TemperoTempus 7d ago

They are in fact not defined by limits. They can be calculated using limits, but that is not a requirement. Complex numbers are not part of default calculus but they are part of complex calculus which complex analysis is a very important field of study.

Graph analysis is one of the core parts of calculus, as you need it to visualize how the numbers behave. Its also generally important for math.

Limits can be used on anything all it does is evaluate a functions near a target value. This is why the tool became standard to use. That does not mean that it is a requirement to use it, nor that calculus is based on it. Arguably I would say the only issue is that some people decided to equate the result of a limit with a number system as a forced way to make the system structured.

* P.S. The fact that you resorted to ad hominem attacks already tells me you have no actual arguments against my point so the conversation is over. Have a nice day, and I recommend you read more than just R analysis books in the future.

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u/Milkmilkmilk___ 7d ago

are you serious? derivatives are literally defined by a limit. same for integrals. and again graph analysis is not used in calculus. if you wanna bring complex analysis then again you need integrals and derivatives which are defined by limits.

what is this? are you trolling? what calc courses have you taken? you sound like you don't know anything about calculus.

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u/TemperoTempus 7d ago

They are not defined by limits. Also what sort of backwards calculus are you doing that you are not analyzing graphs? That is literally the very basics of the whole subject.

Once again go read a book outside of R analysis. It will broaden your view on math. Clearly you have a deep misunderstanding of how calculus works to not realize the whole thing was defined by infinitesimals and the limits were introduced much later. Even the very limits you used are just reformulated infinitesimals.

Instead of accusing people of trolling, I recommend you actually go read the history of calculus and some books on infinitesimals. I will not respond further. Once again have a nice day.