I know that $0/0$ is indeterminate. What I think is (following $ac=b$ as definition for $a/b=c$) $0/0$ can take any value as $0.c$ is always $0$. Hence $c$ could be anything. But consider a function: $$\dfrac{x^2-1}{x-1}$$ At $x=1$ the expression is of the form $0/0$. But the limit is clearly defined at $x=1$(limit is $2$) . So why cannot we define the value of the function to be $2$ at $x=1$? Since $0/0$ is intrinsically indeterminate, we can assume it to be just about anything. But instead of leaving the value of the expression undefined, isn't it better if we let it assume the value of the limit?
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1Key words: "removable discontinuity". – KReiser May 17 '20 at 19:09
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1Yes, you could define it to be that, if you want continuity. You could however define it any other way you please since the expression itself is not defined there. – Allawonder May 17 '20 at 19:57
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See this answer. – PinkyWay May 26 '20 at 12:37
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There are two primary uses of the word "is" in mathematics.
The first is to introduce an adjective, like saying "13 is odd" or "22 is even". This use of "is" can be replaced by "has the property of being", so "13 has the property of being odd".
The second is used to denote equality: we say that "2 + 3 is 5" to mean that the function "+", when applied to the numbers $2$ and $3$, produces a particular value, and that value is exactly $5$. In general, when we define functions, the critical quality that we seek is that if $f(a) = b$ and $f(a) = c$, then $b = c$, i.e., that $f(a)$ denotes, for any number $a$, a single thing. (For functions that take two arguments, like "+", the rule is that $f(a, b) = p$ and $f(a, b) = q$ means that $p = q$.)
What you're suggesting is that we allow $0 / 0$ (the operation of the division function on the two arguments $0$ and $0$) to mean multiple different things. That's something mathematicians could have chosen to do. But if they did so, they'd have to revise the definition of "function", and rewrite all the rules for what it means to be a field (like the real numbers), and a bunch of other things. They decided (collectively, over a long period of time) to not do so. Those other things were deemed more important than assigning a meaning to the string of symbols 0 / 0.
Of course, things like your quotient of polynomials didn't escape anyone's attention, and they've spent a lot of time formalizing the notion of limits, saying what it means for two functions to agree except on a set of measure zero, defining terms like "removable singularity", etc. These notions make precise the kinds of ideas you're interested in, but without breaking everything else that came before.
John Hughes
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The function in question is always equal to $x+1$ provided $x\neq 1$. At $x=1$ (exactly 1, not →1) , we cannot plot a valid point on the graph, just like you said it could be anything. We can however define it to be $2$ at $x=1$ just for the sake of making it continuous.
sai-kartik
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