A layman's motivation for non-standard analysis and generalised limits Disclaimer: My apologies for making such a long question. The question is possibly also rather specific, but I hope that (some parts of) it might be useful in general. 
Background: I have recently learnt some non-standard analysis and generalisations of limits introduced via (ultra-)filters. The way I see things is strongly inspired by Tao's short article on these issues. The rough and basic idea (to make precise which piece of mathematics I am talking about) seems to be that almost all structure on $\mathbb{R}$ can be "imported" to the sequences in $\mathbb{R}^\mathbb{N}$ by the following construction: 


*

*Take an ultrafilter $p$ (an subset of $\mathcal{P}(\mathbb{N})$ that can be thought as the family of all "large" subsets of $\mathbb{N}$). Use this to get a well-behaved notion of $p$-almost all $n$.

*Declare a sentence about real numbers $\phi(x,y,\dots)$ to be $p$-true about sequences $(x_n)_{n=1}^\infty,(y_n)_{n=1}^\infty,\dots$ if and only if $\phi(x_n,y_n,\dots)$ holds for almost all $n$. Notice that $p$-truth same rules as "ordinary" truth.

*Transfer as much structure as you need in accordance to $p$-truth. For instance $x + y = z$ should mean that $x_n + y_n = z_n$ for $p$-almost all $n$. If needs be, divide out the equivalence relation $x \sim x' \Leftrightarrow x_n = x_n' \text{ for $p$-almost all $n$}$. Call $\mathbb{R}^\mathbb{N}$ hyperreals with this structure, and denote it by $*\mathbb{R}$
The key facts about these hyperreals seem to be that:


*

*There is the transfer principle, stating very roughly that same statements are true for reals and hyperreals, as long as you keep them intrinsic (i.e. quantification is only on the underlying set ($\mathbb{R}$ or $*\mathbb{R}$ rather than some external set like $\mathbb{N}$) and not too complex (no quantification over sets, etc.)

*There is the countable saturation stating very roughly that if any finite number of sentences $\phi_1, \phi_2, \dots, \phi_n$ can be simultaneously realised, then all these sentences can be simultaneously realised. 

*There are infinitesimals (like the one given by $x_n = \frac{1}{n}$), infinities (like the one given by $x_n = n$), and ideas such as (uniform) continuity/differentiability etc. admit more elegant definitions and proofs.

*A notion of a fully generalised limit emerges naturally by taking $p\text{-}\!\lim_n x_n = l$ where $l$ is the unique real that is infinitesimally close to $x$ (provided $x$ is bounded).
Being a student of mathematics, I have a rather technical interest in those issues. I have read a number of proofs using ultrafilters (mostly in ergodic theory/number theory, so it did not involve much analysis as such), and I might try to use these concepts working on mathematical problems. 
I am going to present a short talk on the concepts discusses above, and I have learnt that the audience will include many non-mathematicians: there will be physicists, economists, even biologists. Hence the question:
The question: How can one motivate non-standard analysis to someone with just a basic mathematical background? What are the reasons why a beginner mathematician would find it interesting? 
Motivation known to me includes mainly:


*

*Existence of actual infinitesimals/infinities and fully generalised limits.

*Simplified $\varepsilon$-management. (but: only interesting if you actually get your hands dirty with $\varepsilon/\delta$-type calculus)

*Applications of ultrafilters in combinatorics, like Ramsey's theorem for graphs or Arrow's theorem for voting.


Is there more? Are the above valid and convincing? Are there some additional conceptual key properties of $*\mathbb{R}$ that I did not mention?
Additional request: I would also highly appreciate any corrections to the thoughts presented above. They are rather vague by design, but where my thinking is flawed or suboptimal, I will be grateful to anyone who sets it right.
 A: 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)
