Say we have an infinitesimal value ‘dx’ and assign that to a component of some vector (let’s use a 2D vector for simplicity) ‘v = [dx, 0]’
Now let’s rotate that vector on the xy plane by anything not a multiple of 90deg.
We have a vector with components technically less than infinitesimal?
I know anything with an infinite gets into special rules, but my lack of math knowledge made this little paradox a bit hard to munch on.
There are certainly infinitesimals that are larger or smaller than other infinitesimals. In fact, infinitesimals work like any other number when it comes to arithmetic. $dx$ is an infinitesimal and so is $dx/2$, and they satisfy $0 < dx/2 < dx$ as you'd expect. Infinitesimals whose ratio is non-infinitesimal are said to be of the same order. Not all infinitesimals are of the same order. $(dx)^2$ is a higher order infinitesimal than $dx$, and as you'd expect, $0 < (dx)^2 < dx$.
So here we have a vector $dx\hat{\mathbf{x}}$. We rotate it by $\theta$ and get the vector $dx\cos\theta \hat{\mathbf{x}} + dx\sin\theta\hat{\mathbf{y}}$, and its length is $\sqrt{(dx)^2\cos^2\theta+(dx^2)\sin^2\theta} = \sqrt{(dx)^2} = dx$.
Incidentally, using infinitesimals instead of limits to do calculus is called non-standard analysis. Which to a physicist like me is an odd name, as we use it far more often than limits. Non-standard analysis can be put on rigorous grounds and it works about like you'd expect, though there are some subtleties to watch out for.