,

$\mathop {\lim} \limits_{x \to \infty} \frac{1}{x} = 0$

has a concise meaning, defined in terms of the universal (∀) and existential (∃) quantifiers for x ≠ 0:

$\forall \epsilon \in \mathbb{R} \gt 0 \hspace{8pt} \exists \delta \in \mathbb{R} \hspace{8pt} \left| x \right| \gt \delta \Rightarrow \left| \frac{1}{x} \right| \lt \epsilon$

"We can always squeeze the fraction arbitrarily close to zero."

Check (ε, δ)-definition of limit (Wikipedia)

The former is just a different, perhaps more intuitive way, of writing it.

"We can always squeeze the fraction arbitrarily close to zero." — jorndoe

But arbitrarily close to zero is not zero and is never zero. The limit expression is always > zero.

The limit is synonymous to asymptote in the case in question, and an asymptote is never reached. That's why "=" is appropriate.

You know this is accounted for in the definition right? This is how the definition works. 'For all epsilon greater than 0...'

But arbitrarily close to zero is not zero and is never zero. — Devans99

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? :)

I've moved this to the Lounge as there is no philosophy of mathematics in it. The thread is simply about a misunderstanding of the meaning of the limit operator, which the respondents have amply explained.

Are you saying the equal sign means 'arbitrary close' rather than 'equals'?

How about fractals with an infinite circumference and a finite area? — jorndoe

Fractals have a potentially infinite circumference (potentially depending on how many calculations we do). Fractals never have an Actually Infinite circumference.

Are you saying the equal sign means 'arbitrary close' rather than 'equals'? — Devans99

No. The lim, as defined, is zero.

No. The lim, as defined, is zero. — jorndoe

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.

Then it's defined wrong. There is no value of x for which 1/x = 0. — Devans99

So? That's not what it means. Check the link, or some of the other online resources. — 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.

it seems like you're not reading (or understanding) the mathematics and/or definitions. — jorndoe

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.