The problem is to translate the definition of limitation into predicate logic:
For all $\varepsilon > 0$, there is $\delta > 0$ such that when $| x - x _ { 0 } | < \delta , | f ( x ) - f ( x _ { 0 } ) | < \varepsilon $ holds.
My definition is that:
$P(x): x >0.$ $Q(x,x_0,y): |x-x_0| < y.$ $G(x,x_0,y)=|f(x)-f(x_0)|<y.$
$(\forall \varepsilon)(P(\varepsilon) \Rightarrow(\exists \delta)(P(\delta)))$
And I don't know how to go next or if my definition and answer is correct.
Let $X$ be the domain of $f$. We write $\lim_{x\to x_0}f(x)=f(x_0)$ if $$\left(\forall\epsilon\right)\left(\epsilon>0\Longrightarrow\exists\delta\left(\left(\delta>0\right)\wedge\forall x\left(\left(x\in X\wedge|x-x_0|<\delta\right)\Longrightarrow|f(x)-f(x_0)|<\epsilon\right)\right)\right).$$
The rest is just substitution:
$$\left(\forall\epsilon\right)\left(P(\epsilon)\Longrightarrow\exists\delta\left(\left(P(\delta)\right)\wedge\forall x\left(\left(x\in X\wedge Q\left(x,x_0,\delta\right)\right)\Longrightarrow G\left(x,x_0,\epsilon\right)\right)\right)\right).$$