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We all know the formal definition of limit. I'm trying to write a formal definition for "no limits". For context, this is for a very stupid tattoo. Don't drink and bet with tattoo artists, friends.

Here are my four options. Options 1/2 and 3/4 are the same aside from substitution s.t. for colons. Before I get this permanently inked on me with flaming hearts and daggers, I want to get some other eyes on it. Are there any mistakes? Is there a better definition? Which of these do you think is the most accurate?

Option 1: \[ \forall x_0 \in \mathbb R; \; \exists \epsilon > 0: \Big \lvert f(x_0) - \lim_{x \to x_o} f(x) \Big \rvert < \epsilon \; \land \; \not\exists \delta > 0: \Big | x - x_0 \Big | < \delta \]

Option 2: \[ \forall x_0 \in \mathbb R; \; \exists \epsilon > 0 \text{ s.t. } \Big \lvert f(x_0) - \lim_{x \to x_o} f(x) \Big \rvert < \epsilon \; \land \; \not\exists \delta > 0 \text{ s.t. } \Big | x - x_0 \Big | < \delta \]

Option 3: \[ \forall x_0 \in \mathbb R; \; \forall \epsilon > 0; \; \Big \lvert f(x_0) - \lim_{x \to x_o} f(x) \Big \rvert < \epsilon \not \implies \exists \delta > 0: \Big | x - x_0 \Big | < \delta \]

Option 4: \[ \forall x_0 \in \mathbb R; \; \forall \epsilon > 0; \; \Big \lvert f(x_0) - \lim_{x \to x_o} f(x) \Big \rvert < \epsilon \not \implies \exists \delta > 0 \text{ s.t. } \Big | x - x_0 \Big | < \delta \]

(Note, I normally use centernot for does not imply but MathJax doesn't support it)

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    $\begingroup$ I guess I would do $\liminf_{x \to x_0} f(x) < \limsup_{x \to x_0} f(x)$ and give the artist a break :-). $\endgroup$
    – copper.hat
    Commented Jun 23, 2023 at 18:12
  • $\begingroup$ What do you mean with "no limits"? That the absolute value tends to infinity? $\endgroup$
    – md2perpe
    Commented Jun 23, 2023 at 18:26
  • $\begingroup$ Tattoo Artist is likely to mix up $\epsilon$ with $e$ & $\delta$ with $8$ & $x_0$ with $x0$ ETC. Making Corrections will be Difficult. Stick with using $\infty$ to indicate that there are no limits. $\endgroup$
    – Prem
    Commented Jun 23, 2023 at 18:47
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    $\begingroup$ The liminf and limsup version works better too because it's more general whilst still concise: it handles the case of divergent behaviour (to $\pm\infty$) as well as the oscillatory cases $\endgroup$
    – FShrike
    Commented Jun 23, 2023 at 21:20
  • $\begingroup$ ooh, the liming/limsup version is better but less ridiculous. But I think I might go with that. Thanks! $\endgroup$ Commented Jun 26, 2023 at 1:42

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