[lupire]

Your limit definition is the same as the part you quoted, so it's not clear what your question is. I also don't see what you are quoting.

Curvature is inverse of radius.

Decreasing angle is equivalent to increasing radius, and this decreasing curvature. This, as angle decreases, the curve becomes close to a straight line, and that straight line approaches a vertical line.

As usual, 3B1B created a quintessential visualization and explanation.https://m.youtube.com/watch?v=S0_qX4VJhMQ

[enizor2]

I quote the second paragraph of the Derivatives section. (which was edited to a better, but not yet enough, sin(h)≈h and cos(h)≈1 when h is close to zero).

I perfectly understand that around 0, sin(x) ~ x and cos(x) = 1 + o(x) but it isn't obvious geometrically, unlike what the article implies.

From my point of view, increasing radius / decreasing curvature only gets you sin(x) -> 0 ; cos(x) -> 1, but that isn't enough to obtain the derivatives.

I found a geometric proof in [1] but that part is the longest and hardest of the page. I was wondering whether the author found a clearer way to express is.

[1] https://www.mathsisfun.com/calculus/derivatives-trig-proof.h...

EDIT: after looking at 3B1B's video, the "small" triangle d(sinΘ) by dΘ figure would be a better way to explain the derivative, rather than an "not hard to see geometrically" approximation that isn't enough to conclude.