Suppose I write down the Manhattan distance from the origin to a point (x,y) in terms of a series of n steps of length x/n in the x direction, alternated with m steps of length y/m in the y direction: $$d_{Manhattan} = \sum_{i=1}^n \frac{x}{n} + \sum_{i=1}^m \frac{y}{m}$$
Imagine walking an approximately diagonal line towards (x,y), zig-zagging parallel to the x and y axes.
Now, if we take the limits as $n\rightarrow \infty$ and $m\rightarrow \infty$ our path should approach the straight line connecting the origin to (x,y), suggesting that in the limit the Manhattan distance should equal $\sqrt{x^2+y^2}$. Why is this not the case? Is there a way to correctly arrive at Pythagoras by taking a limit using infinitesimal steps along the axis directions?
You can't add up an uncountably many number of immeasurably small points and expect to get the length as a sum.
Yes, the shape and resulting figures both end up being the same line segment but our methods of measuring the length is different. It's a bit like one person measuring the line with a ruler marked in inches saying it is $7$ and another measuring it with a ruler marked in centimeters and saying it is $17.78$.
And lest you think I am being glib, consider we have rulers that measures number of steps. Yours measures number of centimeter steps a tiny gnome would take and you measure that it is $17.78$ centimeter straight steps. Mine measures the number of steps the gnome takes if he goes $1$ centimeter in one direction then turns 90 degrees and goes $1$ centimeter in another. I measure it is $25.15$ centimeter steps. Your ruler is marked of in fractions of steps to infinite precision. So is mine but each of my units is $\frac 1{\sqrt 2}$ as long as each of your units.
We both have the formula. Length = number of steps $\times$ length of steps and an but my number of steps is $\sqrt {2}$ times your number of steps. And for both of us, as length of steps $\to 0$ then number of steps $\to \infty$ and the length of the line as each of us measure it will be $\lim_{\text{length of steps}\to 0}$ number of steps $\times$ length of steps. But for you: number of steps $=\frac {17.78}{\text{length of step}}$ and for me: number of steps $=\frac {\sqrt 2\cdot 17.78}{\text{length of step}}$
If we measure our tiny gnomes with a timer we will notice.... mine is walking slower than yours.
Now notice. Since we can't have measure length as length $=\infty\cdot 0$. But we also have speed = length/time. And we can have instantaneous speed by taking limits. If speed = length/time then length = speed$\cdot$ time. .... And my gnome is slower than yours.
But it's not that my gnome is always slower than yours. Our gnomes live with different metrics. Your gnomes speed is constant at any angle. Mine is not. My gnomes speed matches yours at $0, 90, 180, 270$ degree angles but otherwise is slower than yours depending on the angle with my slowest speed and $45,135, 225, 315$ angles where my gnomes speed is $\frac 1{\sqrt 2}$ of yours.