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


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.


$\infty$ 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 $-\infty$

  • $\begingroup$ 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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