One thing that has been hard to wrap my mind around- take a power function such as $x^2$ or $x^8$. I know the domain of x is infinite, unbounded.
On the one hand such power functions increase in slope with each passing increment - for instance for $x^2$ the derivative is $2x$ so at $x=100$ the slope is $200$, which is a lot, and it only goes up from there. At $x=1000$, the slope is $1000$.
Yet the domain is not bounded. We can picture an x = 3000. The y value must be astronomical. Well it has a value - 9M. It's just hard to mentally conceive that the x values extend to infinity when the y values get so increasingly high.
Has anyone thought about this? Is there any terminology and body of conceptualization around these ideas? On the one hand I can conceive how the graph extends, and on the other hand it is hard to fathom, like the curve of the y is not enough to stop x from breaking free and therefore there is no asymptote, but only just. When I picture a graph of a parabola and zoom out in my mind, it just seems impossible to zoom out enough to picture big values of x. When you zoom out on desmos.com for x^2, the more you zoom out, the more the parabola approximates a line just along the vertical axis. This is like a paradox, something akin to Zenos paradox. Even x^1.1 on desmos seems to curl to a limit and ultimately approach the Y axis. Give it a try.
I don't have a term for this "paradox", but if you have heard of a hyperbola, which I am sure you have, you know that the "sideways parabolic shapes" of the hyperbola approach a limit, which is an asymptote of a straight line, b/a. Applying this to the first derivative as you mentioned, the scale as you shift it out rises much faster than the slope of the first derivative of the function x^n. I have been researching a little bit, and I found out about what you meant with Zeno's Paradox, and I'm not sure what that has to do with this problem, except that if you think about f(x) = x^n with n>1, then you can think about a geometric series. If you don't know what I am talking about, you can find the convergence of a geometric series if the common ratio is less than 1, even if the series is infinite. However, the first derivative, which measures rate of change, similar to the common ratio in a geometric series, is always larger than 1 if n>1, so the function x^n when n>1 never converges to a point, even going on infinitely. x^0.5 would not converge either, as parabolas and square roots are not geometric, but you can think about them in that way. When you zoom out on y = x^0.5, y seems to converge at y = 0. Hope this helps!