Stabilization of lower approximation
Define $h(k,x)$ as the maximal $\ell \leq k$ such that program $x$ halts on inputs $1,\ldots,\ell$ after at most $k$ steps. Clearly $h(k,x)$ is computable. Define $g(k,x) = -1/h(k,x)$. Clearly $g(k,x)$ is computable and monotone.

If $x$ computes a total function then $\lim_{k\to\infty} g(k,x) = 0$. Otherwise, if $s$ is the first input on which $x$ doesn't halt then $\lim_{k\to\infty} g(x,k) = -1/(s-1) &lt; 0$.

Suppose now that some computable function $g'(k,x)$ satisfies $\lim_{k\to\infty} g(k,x) = \lim_{k\to\infty} g'(k,x)$ for all $x$, and $g'(k,x)$ eventually stabilizes for all $x$. Thus $x$ is total iff $\exists k.g'(k,x) = 0$, putting TOT in $\Sigma_1$. However, TOT is known to be $\Pi_2$-complete, and in particular doesn't belong to $\Sigma_1$.