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)