Math Is Fun Forum

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Undefined?

Is

  undefined?

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Undefined?

∞ is not a real number, is it? In fact, it’s not even a number. Is multiplying a number by a non-number defined?

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Undefined?

In fact, it’s not even a number.

There are fields of mathematics which consider it a number.  Game theory is one example.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: Undefined?

The reason I ask:

After solving a differential equation I get a relationship between mass and time:

, where k and C are both constants, k being given as positive.

The question then asks me to "show that this model predicts that the eventual resulting mass of the substance is equal to its initial mass, and state this initial mass".

The initial mass is obviously 40, but I'm not sure how to go about showing that this is equal to the eventual mass (i.e. as t tends towards infinity) due to the two t's in the equation. From my calculator I can see that the exponential expression tends towards 0, but how do I "show" it? The only way I can think of is to say something along the lines of "

" decreases faster than t increases, as t increases" (horribly worded I know). Is that what the question wants?

Thanks.

Last edited by Daniel123 (2008-04-20 09:53:25)

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Undefined?

So that given any epsilon > 0, there exists a t such that:

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Undefined?

If you think of those numbers as limits, then you can make it equal anything (and so it must be undefined).

But replacing 1/x by k/x in that argument lets you conclude any result you like.

(In other news, I have no idea why LaTeX is showing those x-->∞ things in the third line differently.)

Edit: Double post collision.

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Why did the vector cross the road?
It wanted to be normal.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Undefined?

(In other news, I have no idea why LaTeX is showing those x-->∞ things in the third line differently.)

You used the double dollar, that’s why. Try this instead.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Undefined?

Ah, I see. Well, not really, but I understand anyway.
Thanks!

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Why did the vector cross the road?
It wanted to be normal.

moxiuming

Member

Registered: 2008-04-20

Posts: 7

Re: Undefined?

∞ is not a number,so 0*∞ is undefined.if it is defined,what the result will be?since 0 is pulling the result to 0 while ∞ pulling the result to ∞,who will win at last? no one knows. but there is a type of limit who is "0*∞" for example (x-1)/(x^2-4*x+3) where x approaches 1.

TheDude

Member

Registered: 2007-10-23

Posts: 361

Re: Undefined?

The only way I can think of is to say something along the lines of "

" decreases faster than t increases, as t increases" (horribly worded I know). Is that what the question wants?

Yes, that's what it's asking.  Technically it wants you to show that

but it's basically the same thing once the constants are removed.  Just by observing you can see that

will approach 0, since t is growing linearly while e^-t is decreasing exponentionally, but that's not exactly rigorous.  I can't think of any easy theorems you can use to prove this so I'd just go with Ricky's suggestion and make a proof using the definition of limits.  Be sure to include C and k for completeness, they shouldn't make the problem that much more complicated.

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Wrap it in bacon

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: Undefined?

Ok. Thanks everyone.