A layman's motivation for non-standard analysis and generalised limits

I think you're downplaying "simplified $\epsilon$-management": that's one of the main reasons why you want infinitesimals in the first place!

e.g. the formula for the limit

$$ \lim_{x \to a} f(x) = \operatorname{st} f(a + \epsilon) $$

whenever the limit exists and $f$ and $a$ are standard and $\epsilon$ is a nonzero infinitesimal. $\text{st}$ is the "standard part": i.e. rounding a limited number to the nearest standard number. (also, given the conditions on $a$ and $f$, this limit exists if and only if the right hand side has the same value for all nonzero infinitesimals $\epsilon$)

Another example is that there's a really neat argument that any continuous standard function on $[a,b]$ has a maximum. The sketch is:

• "Enumerate" the interval $[a,b]$ by splitting it into (hyper)finitely many intervals
• The set of left endpoints is (hyper)finite, and so among them, $f$ has a maximum at, say, $c$.
• We must have $f(\operatorname{st} c) = \operatorname{st} f(c)$, and it is the maximum of $f(x)$ at standard points.

(and then, transfer the theorem to get the corresponding fact for all continuous functions and closed intervals)