Limits and Infinity (1 Viewer)

[Sy123]

 

[Fizzy_Cyst]

I hate it when it says that the limit IS EQUAL TO infinity. Doesn't make sense to me. Just say it diverges or something.

That's about as much as I can contribute, lol.

Thinking about it, like in terms of the maths which I remember, which is very little, like in terms of asymptotes, which I associate with limits, it's kind of like saying that x converges to infinity, but to get to infinity there must be no bounds.

I'm just confusing myself.

I'm sure there is way more to this, but, thanks for the mindf**k.

 

[seanieg89]

This is true, given the standard way of topologising the extended reals.

(In fact it is trivially true for any topology, but one does need a topology / definition of convergence before such a statement makes sense.)

 

[Sy123]

This is true, given the standard way of topologising the extended reals.

(In fact it is trivially true for any topology, but one does need a topology / definition of convergence before such a statement makes sense.)

Well I was thinking of how one can write as though it is a number

If it were a number, we could do a numerous things that would give us contradictions

For example













So its clear that one cannot treat as some sort of number, rather the way it looks like to me, the statement

has is indefinite, since there is no defined limiting point

Am I wrong in my thinking?

 

[seanieg89]

Well I was thinking of how one can write as though it is a number

If it were a number, we could do a numerous things that would give us contradictions

For example













So its clear that one cannot treat as some sort of number, rather the way it looks like to me, the statement

has is indefinite, since there is no defined limiting point

Am I wrong in my thinking?

Yes, because the notion of a limit on the extended reals need not satisfy all of the properties of taking limits on the reals.

To talk about limits and convergence, all you need is a set and a topology on it.

And your last line further assumes that the extended reals are a field, so we can divide by infinity. There isn't really a sensible field structure on the extended reals though.

Ps, when we do define multiplication on the extended reals, we do in fact have infinity/2=infinity.

 

[Sy123]

Yes, because the notion of a limit on the extended reals need not satisfy all of the properties of taking limits on the reals.

To talk about limits and convergence, all you need is a set and a topology on it.

And your last line further assumes that the extended reals are a field, so we can divide by infinity. There isn't really a sensible field structure on the extended reals though.

Ps, when we do define multiplication on the extended reals, we do in fact have infinity/2=infinity.

From my brief knowledge of infinities, there is also an

Is the infinity that you say is: "the notion of a limit on the extended reals need not satisfy all of the properties of taking limits on the reals."

The same as ?

I always thought to be simply saying, "increase indefinitely", and not actually "infinity" itself

 

[seanieg89]

From my brief knowledge of infinities, there is also an

Is the infinity that you say is: "the notion of a limit on the extended reals need not satisfy all of the properties of taking limits on the reals."

The same as ?

I always thought to be simply saying, "increase indefinitely", and not actually "infinity" itself

Well not quite, the reals have cardinality strictly greater than that of the natural numbers, and hence so do the extended reals.

Well, you can think of it as the notion of increasing indefinitely for the purposes of limit computation, but you can also think of the extended reals as an actual object. Just with the latter point of view, you cannot assume that limits on the extended reals have all of the same properties as limits on the reals.