Indeterminate: the hidden power of 0 divided by 0

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Cites Steven Thomas Hatton · 2019-01 · Mathematics Stack Exchange · on “Is the meaning of *indeterminate* in the context of polynomial theory the same as in the context of, say, L'Hopital's rule?”

This question is a follow-on to Is "indeterminate" a synonym for "variable" or for "transcendent"? [LINK] . I have reproduced the quotations and refined some of my original observation regarding them. See the end of this post for the statement of my question. If you only follow one hyperlink, I suggest it be the one to the video. The following excerpt is from Fundamentals of Mathematics, Volume 1 Foundations of Mathematics: The Real Number System and Algebra, [LINK] Edited by H. Behnke, F. Bachmann, K. Fladt, W. Süss and H. Kunle. (BBFSK) Bold text emphasis added. [W]e have already seen that in general the function $x\to\sum_{i=0}^{n}a_{i}x^{i}$ does not uniquely determine the $a_{i}.$ But f

Cites Lex_i · 2019-07 · Mathematics Stack Exchange · on “Indeterminate Solutions”

I was watching an old video [this video] by Mathologer talking about various problems involving 0 and $\infty$ , but at the end of the video, at roughly 11:40, he concludes ...if you want to make sense of $\frac30$ you also do this by sneaking up on $0$ and of course you know that things explode magnitude-wise...In higher level calculus it actually makes sense to treat infinity like a number and to actually write equations like $\frac30=\infty$ , and you really mean it... $3$ as a number divided by infinity as a number is equal to infinity. He then goes on to say In other branches of mathematics, you sometimes find it actually does make sense to set $\frac00$ equal to 1 He said he would make a vi

Cites Stephan Kolassa · 2020-12 · Mathematics Stack Exchange · on “Elementary example for the indeterminate form $1^\infty$”

I am talking math with a bright middle schooler, who has not even seen logarithms in class yet. (We have successfully introduced logs as the inverse of exponentiation.) She is intrigued by this video [this video] and indeterminate forms. We have discussed how " $1^\infty$ " is really the same as " $0/0$ ". I would now like to discuss an example of " $1^\infty$ ". Unfortunately, every example I can come up with, and everything I find on the internet, uses that $\frac{\ln(1+t)}{t}\to 1$ as $t\to 0$ [LINK] , which is either " a well known fact [LINK] ", or an application of L'Hospital's rule - both of which I find unsatisfactory. Is there a non-trivial example for the " $1^\infty$ " indeterminate fo