Actually might be better to write:

lim x->∞ 1/x ~> 0

That way, we preserve the information that the limit expression always evaluates to > 0.

I think the approximation is built into the concept of limit so you don't need the extra notation

I so wish you and my youngest (19 year old) son could talk.
He is a Sophomore at an Aeronautical University, and he is on his third year, literally 36 straight months of Calculus.
Me? I gave up math in about 5th grade. When they started mixing in letters WITH numbers I knew I was in over my head.
Oh and Math and I are such good buddies that my idea of Purgatory? Is to wake up each day, be taught the fundamentals of Algebra all day long, begin to understand what is being taught only to go to sleep, wake up the next day with absolutely no retention of what was taught yesterday.
Actually that is what happened in High School so.... :rofl:
I admire and stand in awe of those who can work math wonders. :clap:

The problem being if you take the result =0 and use it somewhere else, you have lost the information that it never actually =0, if you see what I mean. That could lead to an error.

So whenever a limit is evaluated, it’s correct to use the approximately equals sign (~) rather than equals. — Devans99

No. Look up the definition of the limit in any modern textbook or online reference. Do not assume that what a mathematical notation "looks like" is what it literally means.

This could explain some of the rather peculiar results in calculus? — Devans99

No.

No. Look up the definition of the limit in any modern textbook or online reference. — SophistiCat

I know the textbook definition, the question the OP poses is: 'is the textbook definition correct?'.

'is the textbook definition correct?' — Devans99

It's a defined concept so the only way that it might be "incorrect" is if someone introduced a concept that fulfilled a similar role that had better computing or explanatory results w/re to describing how functions behave.

The problem being if you take the result =0 and use it somewhere else, you have lost the information that it never actually =0, if you see what I mean. That could lead to an error. — Devans99

If you could provide an example of such an error that would make discussion a little easier. I tend to think several centuries of people successfully using calculus for a variety of applications weighs against there being any errors short of theoretical technicalities.

I know the textbook definition, the question the OP poses is: 'is the textbook definition correct?'. — Devans99

The question in the OP indicates that you don't know or don't understand the textbook definition.

If you could provide an example of such an error that would make discussion a little easier — Mentalusion

What might happen is someone evaluates a limit and then they take the result as precise when it's only approximate. If it then depends critically whether this value is >, < or equals 0, then anything that includes the evaluation of a limit is suspect.

The question in the OP indicates that you don't know or don't understand the textbook definition. — SophistiCat

How so?

What might happen is someone evaluates a limit and then they take the result as precise when it's only approximate — Devans99

If someone did that, they wouldn't understand properly what a limit is and would be trying to get out of it something which it doesn't purport to be able to achieve.

If someone did that, they wouldn't understand properly what a limit is and would be trying to get out of it something which it doesn't purport to be able to achieve — Mentalusion

So you agree we should write:
lim x->∞ 1/x ~ 0
rather than:
lim x->∞ 1/x = 0
?

I think the approximation is built into the concept of limit so you don't need the extra notation — Mentalusion

What you're suggesting isn't necessarily wrong, it's just unnecessary and inefficient. Math already has enough goofy notation for people to keep track off. Why introduce symbols you don't need?

What's the epsilon which is too large for convergence as determined by the standard epsilon-N construction for the series (1/n)?

No idea.

Crack open any textbook on calculus. The concept of the limit is one of the first things that is covered in a typical calculus course, right after the basics of set theory. You need to understand mathematics before you can discuss philosophy of mathematics.

No idea. — Devans99

Yes, that's exactly the problem.

I don't need to understand advanced calculus to point out an obvious notational error.

Perhaps if you understood elementary calculus you would realise it is not an error.

lim x->∞ 1/x = 0
is an error.
For no value of x does 1/x take the value 0.

Limit points don't have to be part of the set of convergent function evaluations towards a point, they just have to be in (metric) topological space underlying them.

Edit: in real analysis this roughly translates to 'sets don't have to contain their supremum or infimum', 0 is the infimum of any increasing sequence of x's which are plugged into f(x)=1/x - the sequence of function evaluations, that the limit exists and is 0 is ensured by the properties of monotonic decreasing sequences in a complete space.

You are avoiding my argument.

1/x = 0
Is false for all x (undefined for 0).

So writing
lim x->∞ 1/x = 0
is definitely wrong

I'm not actually avoiding your argument, wondering why the limit can be said to exist when it is in the closure of a set (a supremum or infimum) rather than simply in the set itself is one of the first concepts you have to understand or teach when you're teaching convergence of sequences. This is literally what the resources I linked for you explain.

Limit points deal with the more general case of convergence in topological spaces, the mathematics of complete metric spaces are what covers this specific example. Study the links and you'll learn something, continue to ignore them and you'll continue spouting rubbish.

Though yes, I agree that what you are talking about is a sticky point for understanding convergent sequences. This does not make it wrong, this means it requires care to grasp.

I don't see anything in the links to justify writing equal when something is plainly not equal.

This means you don't understand the distinction between a limit of a sequence and its elements. You would if you spent more time studying the links. Who knows, it might take more than 18 minutes of study to understand!

18 minutes was chosen because that's how long you took to survey the links. 18 minutes isn't even a complete introductory lecture on the topic, which usually has at least 2 university level math courses devoted to it, and even more involving it... That's hundreds of hours. You tried for 18 minutes.

All I need is a grasp of the english language:

equal
adjective
the same in amount, number, or size

https://dictionary.cambridge.org/dictionary/english/equal

You are defending the indefensible. For something to be equal it has to be the same.

By that reasoning I could criticize literally anything from a purported position of knowledge as long as the position were written in English. :D

Statistical mechanics? So long as it is in English I know what it means, and can criticize it from the almighty authority of the dictionary. The causes of World War II? I don't even need to read a book on the history, but just need to know English and any opinion put forward on the topic can be criticized from my well-rounded knowledge of the language.

Surely you don't want to double down on that principle.


A bit more seriously:

Sometimes disciplines use specialized language because it's more precise and easier to deal with the technical nature of a topic, and knowledge consists of more than knowing the language that it happens to be written in.

Sometimes disciplines use specialized language because it's more precise and easier to deal with the technical nature of a topic, and knowledge consists of more than knowing the language that it happens to be written in — Moliere

But to write equals when something is not equals?

Mathematics should follow logic.

Alright, I'll try one more time --

It's not the "equals" part that you're not understanding, it's the "limit" part.

Maths has a responsibility to make logical sense.

1/x = 0
Is false for all x (undefined for 0).

So writing
lim x->∞ 1/x = 0
is definitely wrong

1/x is always greater than 0 so it could lead to an error downstream.