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.
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