Originally posted by humyThat is a surprisingly popular guess, but it is not a Theory and there is no evidence whatsoever to support it. It certainly doesn't qualify as 'the standard' either.
I know the standard theory is that the universe is finite in size but unbounded ....
Originally posted by humyI do not know if this is what you are doing, but the most common error is to think infinity is a number. It isn't.
I have recently become pretty suspicious of the concept of infinity and I do now very seriously wonder if the whole concept of infinity is total nonsense...
Originally posted by vivifyScientists don't believe it is an actual physical thing. They believe it is a dimension very similar to the space dimensions. But yes, if time is infinite (an unknown at this point) then the same issue would arise, but as I say above, you still would not be able to find two points in time between which an infinite amount of time had passed.
Since scientists believe that time is an actual physical thing, wouldn't that be an example of infinity? Or is time finite, and doomed to end when the universe does?
Originally posted by twhiteheadif
Simply say "f(x) tends to y as x grows larger". Problem solved.
Originally posted by humyjust thought of a solution to that and its so simple I fail to see why I don't see it before!
"there exists a finite x that is such that f(x) ≈ y and there exists no finite z where z is such that it is both z>x and f(z) < f(x) "
-that was my thinking although haven't worked out how to avoid the vague " f(x) ≈ y " bit above which I don't like.
Originally posted by humyI think you will find, if you look into it, that what you are trying to say has already been said and put into the definition of limits. ie infinity itself is not actually used in calculus other than as a symbol representing what you are trying to say. Even by shortened version would need a rigorous definition similar to what you have said, but you don't need to actually say it all every time you want to write a limit. You say it once, then thereafter use the symbols. But no need to invent new symbols, that would merely cause unnecessary confusion.
Hence the reason why I said it in the more cumbersome and complex way of;
"Definition: To say that x tends to zero is to say that x varies in such a way that its numerical value becomes and remains less than any positive number that we may choose, no matter how small."
Originally posted by humyNot sure if its different, but you can find one on Wikipedia:
(if someone can see a better way of expressing that more formally esp to avoid the word "where", please show me here)
Originally posted by twhiteheadArr, I forgot about the requirement that f(x) need not necessarily be well defined (if defined at all) for if x equals p input that gives the limit L output.
Not sure if its different, but you can find one on Wikipedia:
(notice that infinity is just a special case).
Originally posted by humyUnless I am mistaken, your form does not work for oscillating functions that converge to a point, whereas the Wikipedia version does.
adapting that for p = +infinity by making all the necessary adjustments, I can write:
lim x→∞ f(x) = L ⇒ ∀x ∈ ℝ ( ∃z ∈ ℝ : z>x ∧ |f(z) − L| < |f(x) − L|)
(please will somebody correct me if I haven't expressed that conventional notation exactly correctly or made a logical error)