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Another example: Define $f:\mathbb R^+_0 \to \mathbb R$ by $f(x)=\sqrt x$.

Is the statement $f^\prime (0) = \frac{0.5}{\sqrt 0}$ true/false/undefined/something else?

user182601
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    See here - division by zero is always undefined. Of course, $\sqrt{0}=0$. – Dietrich Burde Jan 14 '24 at 10:34
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    It makes no sense to compare undefined objects. "$A=B$" is meaningless , if $A$ and $B$ are not both defined. – Peter Jan 14 '24 at 10:35
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    depends on the use case. the consensus is that the comparison between undefined and anything else is meaningless – whoisit Jan 14 '24 at 10:37
  • @DietrichBurde: I think you misunderstand. My question is not whether $\frac{1}{0}$, $\frac{2}{0}$, and $\frac{0.5}{\sqrt 0}$ are undefined (I agree they are). My question is about whether $\frac{1}{0}=\frac{2}{0}$ is undefined. – user182601 Jan 14 '24 at 10:46
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    Isn't this the same? Both "terms" are undefined, as you say yourself. How can then the "equality" be meaningful? We don't even know what to compare with an equal sign. In the duplicate they discuss $\frac{1}{0}=\frac{1}{1}$, but your case has this problem even "more". – Dietrich Burde Jan 14 '24 at 10:47
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    Well-formed mathematical expressions refer to a value "out there" (e.g., $\frac{1}{2}$ refers to the (unique) real number that gives 1 if you multiply it by 2). Not every string of characters is meaningful in that sense. This holds for expressions such as "$1/0$" or "$7+-/(5$". Since such expressions are meaningless, comparing them for equality does not make any sense ("not even* undefined"). This is like asking "Is it colder outside than at night?" – PhoemueX Jan 14 '24 at 12:06

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