To solve this problem, we need to illustrate the formal definition of a limit, which uses the epsilon-delta definition. Specifically, we need to find the values of $\delta$ that correspond to $\varepsilon = 0.2$ and $\varepsilon = 0.1$ for the given limit:

$\lim_{x \to 3} (x^3 - 3x + 5) = 23$

Step-by-step breakdown:

1. Restating the problem: We know that for the limit $\lim_{x \to 3} (x^3 - 3x + 5) = 23$, we want to find $\delta$ such that for all $x$, if $|x - 3| < \delta$, then:

$|f(x) - 23| < \varepsilon$

where $f(x) = x^3 - 3x + 5$, and $\varepsilon = 0.2$ and $\varepsilon = 0.1$.

2. Rewrite the inequality: For a given $\varepsilon$, we must solve:

$|f(x) - 23| < \varepsilon$

Substitute $f(x) = x^3 - 3x + 5$ and simplify:

$|x^3 - 3x + 5 - 23| = |x^3 - 3x - 18| < \varepsilon$

Now we have:

$|x^3 - 3x - 18| < \varepsilon$