What are the Limits for Infinity in Mathematical Indeterminacy?

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Discussion Overview

The discussion revolves around the concept of limits involving infinity in mathematical indeterminacy, specifically focusing on expressions like 1^∞, ∞^0, and ∞*0. Participants explore the implications of these forms and the necessity of limits in evaluating them.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that expressions involving infinity cannot be treated as standard mathematical operations and must be approached through limits, identifying them as indeterminate forms.
  • Others argue that while multiplying by infinity may not be technically sound, the expressions are used to illustrate a point, emphasizing the constancy of 1 and 0 under multiplication.
  • One participant questions the validity of the original question, suggesting it lacks clarity or context.
  • Another participant provides specific limit evaluations for each expression, noting that different limits can yield different results, highlighting the indeterminate nature of these forms.
  • There is mention of the need for rigorous definitions of infinity to properly evaluate expressions like 1^∞ and 0^0, with some suggesting that 0^0 is not always defined.

Areas of Agreement / Disagreement

Participants express a mix of agreement and disagreement. While there is a consensus on the necessity of limits for evaluating expressions involving infinity, there are competing views on the interpretations and implications of these forms, leading to an unresolved discussion.

Contextual Notes

Limitations include the absence of rigorous definitions for infinity and the potential for varying interpretations of limits, which contribute to the indeterminate nature of the discussed expressions.

MSI
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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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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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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.
 
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 ?
 
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 definition 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.
 
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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