How to formally write this statement on the existence of the limits?

Question

The existence of the limit from one side does not entail the existence of the limit from the other side.

(Source: Schaum's Outline of Calculus, 6th Edition, by F. Ayres, ‎E. Mendelson, p. 57)

How would you, using the precise definitions of the limits, formally write the above statement?

The precise definitions of the left-hand limit and the right-hand limits according to this source are as follows:

Answer 1

Given a function $f:X \rightarrow Y$ and some fixed $x_0 \in X$, the existence of

$\lim_{x \rightarrow x_0^-} f(x)$

does not guarantee the existence of

$\lim_{x \rightarrow x_0^+} f(x)$

and vis versa.

Answer 2

If we denote the statements "$\lim_{x\to a^{+}}f(x)$ exists" and "$\lim_{x\to a^{-}}f(x)$ exists" as they are, then your statement can be symbolized as follows

$$\neg\left(\lim_{x\to a^{+}}f(x)\text{ exists}\Rightarrow\lim_{x\to a^{-}}f(x)\text{ exists}\right)\wedge\neg\left(\lim_{x\to a^{-}}f(x)\text{ exists}\Rightarrow\lim_{x\to a^{+}}f(x)\text{ exists}\right)$$

Equivalently, by DeMorgan's,

$$\neg\left[\left(\lim_{x\to a^{+}}f(x)\text{ exists}\Rightarrow\lim_{x\to a^{-}}f(x)\text{ exists}\right)\vee\left(\lim_{x\to a^{-}}f(x)\text{ exists}\Rightarrow\lim_{x\to a^{+}}f(x)\text{ exists}\right)\right]$$

Edit: We can further reveal the logical structure of the statement by making use of the following definitions:

1. $\lim\limits_{x\to a^{\pm}}f(x)\text{ exists}\iff\left(\exists L\in\mathbb{R}\right)\left(\lim\limits_{x\to a^{\pm}}f(x)=L\right)$
2. $\lim\limits_{x\to a^+}f(x)=L\iff(\forall\varepsilon>0)\{(\exists\delta >0)[(\forall x\in\mathbb{R})(a<x<a+\delta\Rightarrow |f(x)-L|<\varepsilon)]\}$
3. $\lim\limits_{x\to a^-}f(x)=L\iff(\forall\varepsilon>0)\{(\exists\delta >0)[(\forall x\in\mathbb{R})(a-\delta<x<a\Rightarrow |f(x)-L|<\varepsilon)]\}$

Similar definitions hold for left-hand limits.

I won't present the fully expanded statement here; it's way too big and messy.