Calculus
We write:
lim x->? 1/x = 0
But x tends to but never actually reaches infinity, so the right side never actually reaches zero, so surely its more correct to write:
lim x->? 1/x ~ 0
So whenever a limit is evaluated, it’s correct to use the approximately equals sign (~) rather than equals.
This could explain some of the rather peculiar results in calculus?
lim x->? 1/x = 0
But x tends to but never actually reaches infinity, so the right side never actually reaches zero, so surely its more correct to write:
lim x->? 1/x ~ 0
So whenever a limit is evaluated, it’s correct to use the approximately equals sign (~) rather than equals.
This could explain some of the rather peculiar results in calculus?
Comments (42)
lim x->? 1/x ~> 0
That way, we preserve the information that the limit expression always evaluates to > 0.
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:
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.
Quoting Devans99
No.
I know the textbook definition, the question the OP poses is: 'is the textbook definition correct?'.
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.
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.
The question in the OP indicates that you don't know or don't understand the textbook definition.
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.
How so?
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.
So you agree we should write:
lim x->? 1/x ~ 0
rather than:
lim x->? 1/x = 0
?
Quoting 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?
Quoting Devans99
Yes, that's exactly the problem.
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.
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.
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.
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.
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.
But to write equals when something is not equals?
Mathematics should follow logic.
It's not the "equals" part that you're not understanding, it's the "limit" part.
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.
[math]
\mathop {\lim} \limits_{x \to \infty} \frac{1}{x} = 0
[/math]
has a concise meaning, defined in terms of the universal (?) and existential (?) quantifiers for x ? 0:
[math]
\forall \epsilon \in \mathbb{R} > 0 \hspace{8pt} \exists \delta \in \mathbb{R} \hspace{8pt} \left| x \right| > \delta \Rightarrow \left| \frac{1}{x} \right| < \epsilon
[/math]
"We can always squeeze the fraction arbitrarily close to zero."
Check (?, ?)-definition of limit (Wikipedia)
The former is just a different, perhaps more intuitive way, of writing it.
But arbitrarily close to zero is not zero and is never zero. The limit expression is always > zero.
You know this is accounted for in the definition right? This is how the definition works. 'For all epsilon greater than 0...'
So? That's not what it means. Check the link, or some of the other online resources.
As an aside, these sorts of things are used all the time in physics and other areas (derivatives, integration, etc).
I can only guess how much throughout the cool InSight project - congratz to the team.
How about fractals with an infinite circumference and a finite area? :)
Fractals have a potentially infinite circumference (potentially depending on how many calculations we do). Fractals never have an Actually Infinite circumference.
No. The lim, as defined, is zero.
Then it's defined wrong. There is no value of x for which 1/x = 0. Perhaps you are thinking of Actual Infinity?
Actual infinity, if it existed, would be a quantity greater than all other quantities, but:
There is no quantity X such that X > all other quantities because X +1 > X
Further, actual infinity does not follow common sense or mathematical rules:
oo + 1 = oo implies
1 = 0
Nothing in the real world can you add to whilst it remains unchanged. This logical absurdity implies infinity is not a mathematical quantity.
Quoting jorndoe
@Devans99, it seems like you're not reading (or understanding) the mathematics and/or definitions. There are reasonably good online resources, though it may take a bit of reading if you're new to this stuff.
The point is I do not agree with some of the definitions. How exactly is the axiom of infinity from set theory logical? I just disproved the existence of actual infinity twice above (I note you passed on both arguments).
Actual Infinity is just magic and plain impossible/illogical.