Exploring Infinity: Understanding the Answers to Common Questions

  • Thread starter MSI
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In summary, when dealing with operations involving infinity, it is important to use a limit to determine the result. This is because expressions like 1^inf, inf^0, and inf*0 are considered indeterminate forms and cannot be evaluated without the use of a limit. Therefore, the answer for each of these questions depends on the specific limit being used. For example, the limit of 1^x as x approaches infinity is 1, while the limit of inf^x as x approaches 0 is 1. It is important to use a limit when working with infinities to avoid ambiguity and ensure a proper evaluation.
  • #1
MSI
16
1
what is the answer... and why?

* infinity=inf.
questions
a) 1^inf.
b) inf.^0
c) inf.*0

what is the answer for these and why?
 
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  • #2
You can't really use any operators with infinity in that sense.

If you have a limit in the form of (1[tex]\infty[/tex]), [tex]\infty[/tex]^0) or ([tex]\infty* 0[/tex]) like you have then that's an indeterminate form and you would have to use another method of finding what that limit is for that specific case.

There are some more indeterminate forms including 0/0 and [tex]\frac{\infty}{\infty}[/tex]
 
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  • #3
Multiplying by infinity may not be good math technically, but these equations are obviously just trying to make a point.

Doesn't matter how many times you multiply 1 or 0 by itself.

1 x 1 x 1 x 1 x 1 x 1 ... still equals 1

0 x 0 x 0 x 0 x 0 x 0 ... still equals 0

By definition, any number to the 0th power equals 1.
 
  • #4
MSI said:
* infinity=inf.
questions
a) 1^inf.
b) inf.^0
c) inf.*0

what is the answer for these and why?

The question makes no sense...where did you get it ?
 
  • #5
BobG said:
Multiplying by infinity may not be good math technically, but these equations are obviously just trying to make a point.

Doesn't matter how many times you multiply 1 or 0 by itself.

1 x 1 x 1 x 1 x 1 x 1 ... still equals 1

0 x 0 x 0 x 0 x 0 x 0 ... still equals 0

By definition, any number to the 0th power equals 1.

sure, it is obvious that the limit of 1^n as n goes to infinity is 1 and the limit of 0^n as n goes to infinity is 0, but in the abscense of rigourous defintion of infinity which would allow us to evalute 1^inf, etc, it'ds meaninglss to tlak about such things (though 1^inf can be and is used to represent the limit of 1 ^x). 0^0 is not always defined.
 
  • #6
Rule of thumb: Whenever you're working with infinities, use a limit. There's only one answer that way.

For example:
inf/inf could be seen as:
lim[x->inf] (x/inf) = 0
lim[x->inf] (inf/x) = inf
lim[x->inf] (x/x) = 1

and each answer is just as valid. That's why it's inderteminate.

Applying this to your questions:

a)1^inf.
lim[x->inf](1^x) = 1
OR
lim[x->1(-)](x^inf) = 0
lim[x->1(+)]*x^inf) = inf
b)inf^0
lim[x->inf](x^0) = 1
OR
lim[x->0(-)](inf^x) = 0
lim[x->0(+)](inf^x) = inf
c) inf*0
lim[x->inf](x*0) = 0
OR
lim[x->0(-)](inf*x) = -inf
lim[x->0(+)](inf*x) = inf

Notice that the left and right limits are even different in some cases.
Isn't it wonderful?
 
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1. What is the answer?

The answer is a response to a specific question or problem. It is the solution or explanation that is correct or most accurate based on the available evidence or knowledge.

2. Why do we search for the answer?

As scientists, our main goal is to understand the world around us and to solve complex problems. We search for answers to gain knowledge, improve our understanding of natural phenomena, and ultimately make advancements in various fields.

3. Can there be multiple answers?

Yes, there can be multiple answers to a question. In science, there is rarely one definitive answer as new evidence and research can lead to different explanations or solutions. It is important to consider all available information and critically evaluate each answer.

4. How do we determine which answer is correct?

Scientists use the scientific method to determine which answer is the most accurate. This involves making observations, formulating a hypothesis, conducting experiments, and analyzing data. The answer that is supported by the most evidence and is consistent with scientific principles is considered the most correct.

5. Is the answer always final?

No, the answer is not always final. As new evidence and technologies emerge, our understanding of the world can change, and therefore the answer can also change. Scientists are constantly revising and updating their theories and explanations to reflect the most current knowledge.

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