Infinity x Zero: James' Math Problem

[James Has Questions]

I am a high school student, and I am doing extension mathematics, and me and the rest of the class always get into big arguments about number planes. I say that on a basic 2d numberline with a parabola or hyperbola, when x = infinity, y = 1/infinity, which I think is zero, but they think otherwise. And I also think 1/0 equals infinity. If this relationship is true, infinity times zero equals one! I know it all comes down to limits but, please, someone label me victorious or put me out of my misery.

From James

P.S. sorry if this is the wrong newsgroup I wasn't sure.

 

[AlphaNumeric]

You cannot just plug in infinity and zero into certain equations like y = 1/x and then expect it to obey the 'normal rules'. For a start, infinity isn't a member of the Real numbers, it's a member of the extended Reals.

When you're doing this kind of thing you need to instead use 'limits'. Rather than just put in x=infinity, you consider what happens when x gets really big (or sometimes when x gets really small).

For instance, what is $$y = \frac{3x+4}{x-2}$$ at x=infinity? If you just wack in x=infinty you might give the naive answer y=1. However, that would be wrong. With some thought you can do the following

$$y = \frac{3x+4}{x-2} = \frac{3+\frac{4}{x}}{1-\frac{2}{x}}$$

Now as x-> infinity, the 4/x and 2/x parts go to zero and you end up with y=3.

Because of this requirement to 'approach infinity' rather than just put in infinity, you can't say things like 1/0 = infinity means 1 = 0*infinity. I could just double each side to get 2 = 0*infinity, so that means 1=2. Clearly this is wrong and is a sign you've assumed you can do a certain operation you shouldn't.

It is true that in the limit as x->0, 1/x goes to infinity, but I'd be extremely weary of putting equals signs there without mentioning limits or you'll end up proving 1=2.

What exactly do your friends think the value of 1/infinity is anyway? By using limits (so it's rigorous) you can show that if they think it's any number other than zero, they are wrong. For instance, suppose they think it's k, where k is really really small.

So you've y = k on the y=1/x line. Just pick x to be bigger than 2/k, and you'll see that the limit is less than their guess, so they are wrong.

 

[matt grime]

The first thing to remember is that the number line does not have infinity on it. So 1/0 or 1/infinity are not questions about the number line. Let's switch to calling the number line the Real numbers which we will denote by R.

So, this is not actually a question about R. We have to 'add' something to R to start to make sense of it. The thing we commonly add is a symbol $\infty$ and it is true that 1/0=$\infty$ and 1/$\infty$=0 are definitions we make, but we do not define 0*(1/0) even in this extended set of numbers (just as we do not define 1/0 in R).

Where do these choices come from? They do indeed come from taking limits of things like 1/x as x tends to zero: this is can be made arbitrarily large by taking x arbitrarily close to zero. So we add the limit in and get a new number larger than all real numbers.

Sometimes people also extend the real numbers to include -$\infty$, sometimes people don't: it depends on what you want to do with things.

In any case, (1/0)*0 can't be defined to make sense.

 

[James Has Questions]

Ok

Wow! Thanks guys, I'll show this to my friends and we can all look back and laugh. But in the in graph, just say, y=1/x, when x=infinity would y=0? or is 0 a discontinuity to everything even ifinity. But I suppose, you wouldn't write that in a textbook because infinity is not a real number, so it wouldn't really count.

 

[arildno]

x=0 is not in the domain of the function f(x)=1/x, nor can we extend f to some continuous function F where x=0 IS in F's domain.

 

[matt grime]

when x=infinity

as has been pointed out twice infinity is not an element of the real numbers (the number line, in your language).

 

[James Has Questions]

Alright thanks

 

[robert Ihnot]

Just us L'Hospital's Rule, giving 3/1.