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By slope I mean derivative. Is something being infinity the same thing as being undefined?

Is the slope of a vertical line negative infinity or positive infinity?

  • We would describe it as undefined, because a slope of $-\infty$ would also be vertical. – Chickenmancer Sep 16 '17 at 20:26
  • We might call it infinite if we treat $+\infty$ and $-\infty$ as the same (a projective space). – Michael Burr Sep 16 '17 at 20:27
  • Would we say that the slope of a vertical line approaches negative infinity or positive infinity depending on how you rotate it to make it vertical? –  Sep 16 '17 at 20:27
  • A vertical line is not a function and it cannot have a derivative. If you describe the function of x with respect to y, then sure the derivative is ${dx\over dy} = 0$. – cr001 Sep 16 '17 at 20:27
  • I think the right hand derivative and left hand derivative aren't equal so you can't take derivatives of the vertical line? –  Sep 16 '17 at 20:28
  • can you take the derivative with respect to the equation y=4? Is y=4 a function? –  Sep 16 '17 at 20:30
  • @cr001 Why should functions be the only things that have derivatives? (Curiosity prompts me to ask this) – Simply Beautiful Art Sep 16 '17 at 20:31
  • Yes $y=4$ is a function and it is horizontal. However the vertical line $x=4$ is not a function. – cr001 Sep 16 '17 at 20:31
  • You could call it infinity, but infinity is of course not a number. Anything you can't define a number to is undefined. – Kaynex Sep 16 '17 at 20:32
  • @Simply Beautiful Art By definition of derivative. – cr001 Sep 16 '17 at 20:32
  • The slope must be finite. A vertical line cannot be described via the slope. But it can easily be described with $x=c$ – Peter Sep 16 '17 at 20:35
  • y=1/x is locally at some limiting cases linear approximately the vertical line x=c. Do derivatives not exist there? –  Sep 16 '17 at 20:39
  • @Got No.$\text{}$ – Simply Beautiful Art Sep 16 '17 at 20:41
  • how do you prove that y=4 is a function and x=4 is not a function? –  Sep 16 '17 at 20:47
  • For every $x$-value , we can have at most one $y$-value, otheriwise we only have a relation. The case $x=4$ is an extreme example of such a relation not being a function. – Peter Sep 16 '17 at 20:49
  • are you assuming y=axis is the codomain and x-axis is the domain? If we assumed y-axis is the domain and x-axis the the codomain, you'd be wrong? but domain and codomain are arbitrary –  Sep 16 '17 at 20:51
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    @Got If you want it this way: the derivative is measured as "units codomain per units domain". If the $y$-axis is your domain, then a vertical line has zero slope. – M. Winter Sep 16 '17 at 20:54
  • in formal terms in a relation R defined from $X \rightarrow Y, \forall x \in X, \exists$ at most one y $ \in Y$ and let $ X$ be the x-axis and the $ Y$ be the y-axis,,, and this relation R is a function by definition. –  Sep 16 '17 at 20:55
  • Yes, I assumed that we have $f(x)=\cdots$ which we can also interprete as $y=\cdots$. Of course, we could define $f(y)=4$ but in the univariate case, we usually map $x$-values to $y$-values. – Peter Sep 16 '17 at 20:55

1 Answers1

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You can only compute derivatives of functions $f:\Bbb R\to\Bbb R$ (at least in this context here). A vertical line is no such function. So one can consider it as undefined. At least as long as you insist in defining "slope" as derivative.

But infinities are not the same as something being undefined. This depends on the context.

M. Winter
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