Indeterminate form
- Thread starter burt
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One way to think about [MATH]\infty[/MATH] is the projectively extended reals.
In that system, [MATH]\infty[/MATH] is defined as [MATH]\infty = \dfrac{1}{0}.[/MATH]
All operations involving real numbers except 0/0 are determinate (including r/0 if r is not zero). Most arithmetic operations involving [MATH]\infty[/MATH] and a real number are also determinate, but operations that involve only infinity are indeterminate.
en.wikipedia.org
In that system, [MATH]\infty[/MATH] is defined as [MATH]\infty = \dfrac{1}{0}.[/MATH]
All operations involving real numbers except 0/0 are determinate (including r/0 if r is not zero). Most arithmetic operations involving [MATH]\infty[/MATH] and a real number are also determinate, but operations that involve only infinity are indeterminate.
Projectively extended real line - Wikipedia
Steven G
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No! You should know what \(\displaystyle \dfrac{0}{\infty}\)equals
pka
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Here are mine.
If that is the case for real numbers. then \(\displaystyle 0\cdot \infty=1\) Does anyone really want that?
There is a perfectly worked out, logically consistent, system known as non-standard analysis.
Here is a completely free textbook available for download. Elementary Calculus: An Infinitesimal Approach In Elementary Calculus: An Infinitesimal Approach by Jerome Keisler in chapter 3, there is good proof of this. The chapters and whole book is a free down-load at http://www.math.wisc.edu/~keisler/.
Down load that text. See how enlarging the real number system can accomplish what you seek in a completely logical way.