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[Heron TSG]

meaning the limit does not exist

Why doesn't the limit exist, though? Is the limit not equal to every number, as the line would only be continuous if there were a vertical line between the infinities? This explanation sounds a lot like having an undefined limit, in that you can't define every number.

[Jim Groovester]

meaning the limit does not exist

Why doesn't the limit exist, though? Is the limit not equal to every number, as the line would only be continuous if there were a vertical line between the infinities? This explanation sounds a lot like having an undefined limit, in that you can't define every number.

No.

A limit cannot be equivalent to multiple values at once. That means it doesn't exist.

If you're still not satisfied, try finding the limit at x = 2 for this piecewise function.

f(x) = { 3, x < 2
           5, x > 2
           7,  x = 2

Remember, for a limit to exist, it's left hand and right hand limit must be the same.

For the sake of helping you figure it out, here's an example of a function that has an undefined limit:

f(x) = 1/x²

The limit at x = 0 exists and is undefined, because the left hand limit and right hand limit approach the same location, positive infinity.

In the problem you were marked wrong on, they did not approach the same location, so the limit does not exist. You absolutely can not call a limit that does not exist undefined.

[qwertyuiopas]

Spoiler: Response to early post (if only I had a time machine) (click to show/hide)

(Underline added)

Familiarize yourself with the powers of base 2--I compute them for fun when I'm bored (gotten up to 100 before).

Really? I remember doing  something like that, long ago. Some neat patterns emerge in the 9s and 0s when they are arranged an a right-alighed column. Also interesting symmetry in the carry of each column.
(Unfortunately I can no longer ignore teachers for such things, teachers actually teach new stuff and grade harder in the last few years of high school, and it's a waste of money to try in University.)

[Vector]

Spoiler: Response to early post (if only I had a time machine) (click to show/hide)

(Underline added)

Familiarize yourself with the powers of base 2--I compute them for fun when I'm bored (gotten up to 100 before).

Really? I remember doing  something like that, long ago. Some neat patterns emerge in the 9s and 0s when they are arranged an a right-alighed column. Also interesting symmetry in the carry of each column.
(Unfortunately I can no longer ignore teachers for such things, teachers actually teach new stuff and grade harder in the last few years of high school, and it's a waste of money to try in University.)

Eh, I was playing D&D and a bunch of people were off doing something I wasn't very involved in--so I just took half an hour or so to sit and compute powers of 2.  Didn't discover anything particularly interesting, since it was just a mechanical exercise, wasn't exactly neat, and my brain was kind of fried, but it sure helped later.

Guess great minds run in the same gutter =)

[RedWarrior0]

So I was bored and well, I was wondering something. If f(x) = ax + b, and f^-1(x) = (x - b)/a, what's f²(x)

[ed boy]

That depends on whether you mean (f(x))² or f(f(x)).

If the former, a²x² +b² +2abx.
If the latter, a²x+ab+b.

I suspect the latter from your notation of the inverse function, though.

[RedWarrior0]

Yeah, I figured it out pretty quickly after the post. It's the latter.

[Armok]

"Is true is and only if it is false and is false if and only if it is false" Is true is and only if it is false and is false if and only if it is false.

Is there anything interesting about this statement? What is the minimum requirements to formalize and prove it?

[Virex]

Well, probably start with wording it right, because I think you've replaced some if's with is's...

[Vector]

"Is true is and only if it is false and is false if and only if it is false" Is true is and only if it is false and is false if and only if it is false.

Is there anything interesting about this statement? What is the minimum requirements to formalize and prove it?

"True if and only if it is false and false if and only if it is false" translates to "always both true and false" (via, IIRC, hypothetical syllogism and conjunction elimination).

Thus the first statement is false, but the entire statement has no defined truth value.

[Armok]

Typo, damn, I meant:

"Is true if and only if it is true and is false if and only if it is false" Is true if and only if it is true and is false if and only if it is false.

[Vector]

Typo, damn, I meant:

"Is true if and only if it is true and is false if and only if it is false" Is true if and only if it is true and is false if and only if it is false.

Oh, then that's just a garden-variety tautology.  We can put the quoted statement as P, and then ... the entire thing is still equivalent to P.

[dragnar]

True if and only if it is true, and false if any only if it is false... That has absolutely no meaning. It is true of ANY statement, not just the paradox it is referring to, simply because that is pretty much the definition of true and false: two mutually exclusive states.

I think the whole statement would be written like this: ((a <-> a) ^ (a <-> ~a)) -> ((a <-> a) ^ (a <-> ~a))
The statement is always true, because it is essentially just saying "This impossibility is true if it is true".

[Armok]

What is the minimum axioms needed for it?

[Vector]

What is the minimum axioms needed for it?

Go get yourself a textbook on model theory =)