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Let's start with the equation $$y =\frac 1{(x-1)}$$. 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 $+∞$?

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  • $\begingroup$ Would it be $-\infty$ or $+\infty$ and why? $\endgroup$ – Git Gud Feb 23 '13 at 23:48
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    $\begingroup$ Because the limit from the right is $\infty$ and the limit from the left is $-\infty$. $\endgroup$ – David Mitra Feb 23 '13 at 23:48
  • $\begingroup$ Right. I forgot part of the equation. I'll update my question momentarily $\endgroup$ – user60837 Feb 23 '13 at 23:51
  • $\begingroup$ 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 $\infty$. $\endgroup$ – Michael Hardy Feb 23 '13 at 23:55
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    $\begingroup$ No, infinity and undefined are two different things. $\endgroup$ – CogitoErgoCogitoSum Feb 24 '13 at 0:34
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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:

  1. Is infinity a number?
  2. What exactly is infinity?

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.

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  • $\begingroup$ My apologizes, I forgot part of the equation. It was supposed to be the absolute value of x. $\endgroup$ – user60837 Feb 23 '13 at 23:55
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    $\begingroup$ I edited my answer. $\endgroup$ – Asaf Karagila Feb 23 '13 at 23:56
  • $\begingroup$ 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? $\endgroup$ – user60837 Feb 24 '13 at 0:07
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    $\begingroup$ 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. $\endgroup$ – user60837 Feb 24 '13 at 0:18
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    $\begingroup$ No problem! :-) $\endgroup$ – Asaf Karagila Feb 24 '13 at 0:19