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Say like $\lim_{x\to0}\frac1{x^2}$. My professor said the limit of a function cannot be infinity, and according to my textbook “limit is a finite number” is clearly stated. Infinity, from my knowledge, is neither finite nor a number.

However I do see many people write down infinity as answer for questions like $\lim_{x\to0}\frac1{x^2}$.

I am confused. Which one is correct?

Kelvin Man
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    If the limit is infinite, then the limit doesn't exist. And $\lim_{x\to 0}\frac{1}{x^2}=+\infty $ is correct. – Surb Sep 14 '22 at 11:55
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    When we write $\lim\limits_{x\to 0}\left(\frac{1}{x^2}\right) = \infty$ that is not literally saying that the limit exists and is equal to infinity. It is shorthand for saying "as $x$ approaches zero the value of $\frac{1}{x^2}$ increases without bound" – JMoravitz Sep 14 '22 at 11:56
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    It is worth pointing out as well that there do exist contexts like the extended reals or the Riemann sphere that we could be working in where we do treat $\infty$ like a number (albeit with a great deal of care) in which case we do literally have the limit existing and being equal to infinity. It is perhaps because of that why we use the notation we do for the usual scenario as well. – JMoravitz Sep 14 '22 at 12:01
  • JMoravitz+1 The limit of a function can be infinity. A limit is not necessarily a finite number. If the limit is infinite, then the limit does "exist" (as an element of $[-\infty, +\infty]$). – Anne Bauval Sep 14 '22 at 12:10
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    As you can see from these comments, there is disagreement/ambiguity regarding the "existence" status of $\infty$. But one thing we can all agree on: $\infty$ is *not a number; and $\lim_{x \to 0} 1/x^2$ does not exist* in the real numbers. – Lee Mosher Sep 14 '22 at 12:13
  • Nonetheless, one can formulate a rigorous definition of the expression $\lim_{x \to 0} 1/x^2 = \infty$, as hinted at in the answer of @JMoravitz. – Lee Mosher Sep 14 '22 at 12:15
  • Read here: https://math.stackexchange.com/questions/4469433/probably-silly-question-on-notation – insipidintegrator Sep 14 '22 at 12:44
  • These statements should have been made with some qualification, such as "in this course" for the professor and "in this book" for the textbook. The textbook, especially, should have had a footnote about this issue. Another example of a restriction that can be changed in a later course is the following. In precalculus and beginning calculus courses the domain of $f(x) = \sqrt x$ (when not explicitly specified) is $x \geq 0.$ However, in complex variables courses the domain of $f(x) = \sqrt x$ (when not explicitly specified) is all complex numbers (including negative real numbers). – Dave L. Renfro Sep 14 '22 at 13:09
  • We should simply accept that the limit , strictly speaking , does not exist. I see no merit in extended real lines and similar trials to give "$\infty$" a meaning. – Peter Sep 15 '22 at 09:19
  • @Peter: Infinite limits are used in many investigations in real and complex analysis and, for what it's worth, in these investigations it's often not necessary to view $\infty$ as some kind of number that one might want to arithmetically work with. Many times, at least for real-valued functions, in stating results of interest it is enough to extend the usual order on the reals so that for each real number $r$ we have $-\infty < r < +\infty$ (and, of course, $-\infty < +\infty).$ – Dave L. Renfro Sep 15 '22 at 14:37

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