Infinity and variable

[scoutfai]

I was wondering the following two situations:

1) If x is a real variable, then, is it correct, or is it acceptable rigorously to assign x
x = +infinity
?

2) if x is not necessary a real variable, then is it correct, or is it acceptable rigorously to assign x
x = +infinity
?

I am guessing the first one is wrong, because if x is an element of the real number set, then infinity is not an element of real number, so an unknown x can never be written to be equal to infinity.
I think the second statement is correct, because if x is not necessary an element of real number set, then it is correct to write unknown x is equals to infinity.

 

[torquil]

As you suspected, 1) is incorrect.

2) can be correct. E.g, if the expression is meant as an equation or definition in the one-point compactification or the reals, then it is OK. In that set of numbers, infinity is just another element (one defines an extra element called something like "the point at infinity"), and therefore your x can represent that element if you want it to.

But it is important to specify that you are then working not with the reals, but with a more exotic set.

Here is more on the one-point compactification of R. Read the "An example" section here:
http://en.wikipedia.org/wiki/Compactification_%28mathematics%29" [Broken]

 

[scoutfai]

As you suspected, 1) is incorrect.

2) can be correct. E.g, if the expression is meant as an equation or definition in the one-point compactification or the reals, then it is OK. In that set of numbers, infinity is just another element (one defines an extra element called something like "the point at infinity"), and therefore your x can represent that element if you want it to.

But it is important to specify that you are then working not with the reals, but with a more exotic set.

Here is more on the one-point compactification of R. Read the "An example" section here:
http://en.wikipedia.org/wiki/Compactification_%28mathematics%29" [Broken]

A new concept and terminology to me! Thank you for sharing it.

So my guess on the 1st question is right that it is incorrect to assign x as infinity if x is a real variable.
but what if x is a real variable, and someone writes:
x = lim _ (a -> inf) a

then, is it correct to write like this?
"Yes because limit was used, despite x is defined as a real variable."
OR
"No because x is defined as real variable and a limit approaching infinity is not a real number."
Any thought?

 

[TylerH]

It is correct to say x becomes infinite, but you can't say is equals or approaches infinity. As infinity is a concept to say that it grows infinitely, not a number.

To evaluate the limit you posed, you would notice that it grows infinitely, so you can say $$lim_{a \rightarrow \infty}=\infty$$ to signify that there is *no real limit*, but that it grows infinitely.

 

[disregardthat]

It is correct to say x becomes infinite, but you can't say is equals or approaches infinity. As infinity is a concept to say that it grows infinitely, not a number.

But we do incorporate the symbol $$\infty$$ as a symbol in our mathematical notation in the one-point compactification of the complex plane, or in the extended reals. In this context we have well-defined algebraic rules for manipulating it, it behaves much like a number.

 

[torquil]

A new concept and terminology to me! Thank you for sharing it.

So my guess on the 1st question is right that it is incorrect to assign x as infinity if x is a real variable.
but what if x is a real variable, and someone writes:
x = lim _ (a -> inf) a

then, is it correct to write like this?
"Yes because limit was used, despite x is defined as a real variable."
OR
"No because x is defined as real variable and a limit approaching infinity is not a real number."
Any thought?

My thoughts are:

One can try to determine if the limit lim _ (a -> inf) exists within the reals R. In this case, the answer would be that the limit does not exist. If it had existed (for some other expression), only then would I have written an expression of the form you wrote involving x.

My reason is that when I want to define a quantity x in R, or write an equation for an unknown value x in R, I want the rest of the expression that defines it to be well-defined in R.

But in the wild, different notations are used and problems rarely arise. If someone writes that the limit is equal to infinity, you immediately know that the limit does not exist in R, and that the value within the limiting expressin grows without bounds in the positive direction as a grows. So it is quite convenient.

 

[scoutfai]

It is correct to say x becomes infinite, but you can't say is equals or approaches infinity. As infinity is a concept to say that it grows infinitely, not a number.

To evaluate the limit you posed, you would notice that it grows infinitely, so you can say $$lim_{a \rightarrow \infty}=\infty$$ to signify that there is *no real limit*, but that it grows infinitely.

You wrote :
"x becomes infinite"
"approaches infinity"

Perhaps I get you wrong, but I feel like you are trying to say that infinite and infinity are two different things, can you elaborate more on this aspect?
As far as I knew, all my lecturers and teachers use these two terms interchangeably. Or maybe they really mean two different things, just that I unable to differentiate them?

 

[scoutfai]

My thoughts are:

One can try to determine if the limit lim _ (a -> inf) exists within the reals R. In this case, the answer would be that the limit does not exist. If it had existed (for some other expression), only then would I have written an expression of the form you wrote involving x.

My reason is that when I want to define a quantity x in R, or write an equation for an unknown value x in R, I want the rest of the expression that defines it to be well-defined in R.

Very precise statement, I agree with you.

But in the wild, different notations are used and problems rarely arise. If someone writes that the limit is equal to infinity, you immediately know that the limit does not exist in R, and that the value within the limiting expressin grows without bounds in the positive direction as a grows. So it is quite convenient.

This actually raises another question of mine. Will you agree to say that, in the rigorous, precise way of doing mathematics, one is wrong to write
$$lim_{a\rightarrow\infty}a=\infty$$
but is correct to write
$$lim_{a\rightarrow\infty}a\rightarrow\infty$$

The former is an equality, where is the later is an approaching.
Or both are mathematically correct, precise, rigorous and well-defined?

 

[TylerH]

You wrote :
"x becomes infinite"
"approaches infinity"

Perhaps I get you wrong, but I feel like you are trying to say that infinite and infinity are two different things, can you elaborate more on this aspect?
As far as I knew, all my lecturers and teachers use these two terms interchangeably. Or maybe they really mean two different things, just that I unable to differentiate them?

You assumption is correct. They are logically equivalent. To clarify, they are both correct. But, "x becomes infinite" is more correct. It's a matter of language. A limit can approach a number, but not a concept. When considering the reals, as you have specified, infinity is only a concept. You can write x->infinity, but it is read as "x becomes infinite." No one will be confused if you say "approaches infinity," but the smart alec in the room may nitpick.

 

[torquil]

Very precise statement, I agree with you.


This actually raises another question of mine. Will you agree to say that, in the rigorous, precise way of doing mathematics, one is wrong to write
$$lim_{a\rightarrow\infty}a=\infty$$
but is correct to write
$$lim_{a\rightarrow\infty}a\rightarrow\infty$$

The former is an equality, where is the later is an approaching.
Or both are mathematically correct, precise, rigorous and well-defined?

That funny, 'cause I was just thinking about the same thing when I wrote my answer. Perhaps the arrow would be a good notation. I haven't seen it in the context of using the $$\lim$$, but this is quite common (I think, I'm not a professional mathematician):

$$a \overset{a\rightarrow\infty}{\rightarrow} \infty$$

 

[scoutfai]

That funny, 'cause I was just thinking about the same thing when I wrote my answer. Perhaps the arrow would be a good notation. I haven't seen it in the context of using the $$\lim$$, but this is quite common (I think, I'm not a professional mathematician):

$$a \overset{a\rightarrow\infty}{\rightarrow} \infty$$

What a coincidence that we both thinking the same issue.

So my understanding from your reply is, you deem that the following expression
$$lim_{a\rightarrow\infty} a\rightarrow\infty$$
is more correct than the following expression
$$lim_{a\rightarrow\infty}a=\infty$$
however you aren't sure about it?

I will think the arrow way of writing is more correct than the equal sign, but I am not sure about it also.

 

[torquil]

So my understanding from your reply is, you deem that the following expression
$$lim_{a\rightarrow\infty} a\rightarrow\infty$$
is more correct than the following expression
$$lim_{a\rightarrow\infty}a=\infty$$
however you aren't sure about it?

I will think the arrow way of writing is more correct than the equal sign, but I am not sure about it also.

I agree with you that one is more "notationally correct" than the other. At the same time, I think it is unlikely that anyone would misunderstand any of the two expressions.

 

[disregardthat]

$$\lim_{x \to a} f(x)$$ is either a number or nonsense. There is nothing notationally wrong in saying $$\lim_{x \to a} f(x) = \infty$$. It is merely conventional, most common, and extends the notation of existing limits like e.g. $$\lim_{n \to \infty} 1/n = 0$$, and coincides with notation for broader number systems where $$\infty$$ is included, such as the extended reals, or the one point compactification of C, where unbounded increasing (or absolutely increasing) sequences does have the limit $$\infty$$.

All this considered it would be awkward to insist on notation such as $$\lim_{x \to a} f(x) \to \infty$$. What you probably are confusing this shorthand expression with is "$$f(x) \to \infty$$ when $$x \to a$$".