Is Infinity =Undefined?

Question

Let's start with the equation $$\begin{equation}y =\frac 1{(x-1)} \end{equation}$$ where the positive and negative limit of $x$ at $1$ both approach $+∞$, but at $x = 1$, $y$ is undefined.

I know this is because the denominator of the equation resolves to $0$, but why does $y$ become undefined instead of $+∞$?

Because the limit from the right is $\infty$ and the limit from the left is $-\infty$. — David Mitra, Feb 23 '13 at 23:48
Right. I forgot part of the equation. I'll update my question momentarily — , Feb 23 '13 at 23:51
Putting a single $\infty$ at both ends of the line rather than $\pm\infty$ makes sense when dealing with both domains and ranges of rational functions, and with ranges of trigonometric functions. Then rational functions are continuous everywhere and trigonometric functions are continuous except at $\infty$. — Michael Hardy, Feb 23 '13 at 23:55
@GitGud : You'll probably need to clarify your question further before I can say anything about it. — Michael Hardy, Feb 24 '13 at 00:02
@MichaelHardy Wouldn't one want to have $-\infty \leq x \leq \infty$, for all $x\in \Bbb R$? — Git Gud, Feb 24 '13 at 00:04
In this context, I suppose so, yes, it is a thing. As much so as infinity, but not the same as infinity. — CogitoErgoCogitoSum, Feb 01 '16 at 05:12

score 5 · Accepted Answer · edited Apr 13 '17 at 12:20

5

First of all, infinity is not a real number so actually dividing something by zero is undefined. In calculus $\infty$ is an informal notion of something "larger than any finite number", but it's not a well-defined number.

You might want to read the following:

Edit: the following refers to an earlier version of the question.

Secondly, as the comments remarked, $-\infty\neq+\infty$ when talking about the real line. Note that when $x<1$ we have that $x-1<0$, and when $x>1$ we have $x-1>0$. Therefore: $$\lim_{x\to1^-}\frac1{x-1}=-\infty\\\lim_{x\to1^+}\frac1{x-1}=+\infty$$

The limit is defined if and only if the two sided limits are equal. They are not, so the limit is undefined.

edited Apr 13 '17 at 12:20

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1

answered Feb 23 '13 at 23:53

Asaf Karagila

393,674

My apologizes, I forgot part of the equation. It was supposed to be the absolute value of x. – Feb 23 '13 at 23:55
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I edited my answer. – Asaf Karagila Feb 23 '13 at 23:56
So ∞ is the representation of boundless growth? If this is the case, why isn't it y = ∞ meaning y is boundless? Why is it y = undefined? – Feb 24 '13 at 00:07
Because the function is not defined to be from $\Bbb R$ to $\Bbb R\cup{-\infty,+\infty}$, but rather into $\Bbb R$. So if the value is not a real number (and it cannot be) it is undefined. – Asaf Karagila Feb 24 '13 at 00:12
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That makes sense. Infinity isn't in the set of all real numbers because it's not a defined and tangible number. Thank you for shedding light on this mystery that's haunted me since Algebra. – Feb 24 '13 at 00:18
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No problem! :-) – Asaf Karagila Feb 24 '13 at 00:19

Is Infinity =Undefined?

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