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I'm doing some L'Hopital practice and I came across this example:

https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/lhopitaldirectory/solution17.html

Now, it's perfectly clear how the problem is solved, the one thing that threw me off is that at the end of the answer it says "(Limit does not exist)" and I do not understand why.

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$\infty$ is not a real number, so if you are working in the standard reals the limit does not exist. It can be useful to be more specific and define a limit of $+\infty$ to mean that for any $N$ I give you, you can find an $x_0$ so that $x \gt x_0 \implies f(x) \gt N$ (compare with the $\epsilon - \delta $ definition of a limit). Then when you say a limit is $+\infty$ we know the value doesn't just bounce around or head off to $-\infty$

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  • $\begingroup$ would this be the same as saying that $\lim_{x\to 0} \frac{1}{\left(x\right)^2} = +∞$ does not exist? $\endgroup$
    – Jose V
    May 7 '16 at 14:59
  • $\begingroup$ another comment in case you didn't see when i edited the first one, thanks! $\endgroup$
    – Jose V
    May 7 '16 at 15:06
  • $\begingroup$ It is exactly the same thing. My second point was that it can help to distinguish that behavior from $\lim_{x \to 0} \frac 1x$, which goes positive on one side of zero and negative on the other and from $\lim_{x \to 0} \sin \frac1x$ which wiggles around a lot. $\endgroup$ May 7 '16 at 15:10

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