Limits

So, what are limits? Let’s play with an idea.

Imagine we have a function, any function. Take two points on its curve and draw a line through them — that’s a secant line. Cool, right? Now, what can we do with this line?

Let’s shake the tree a bit. Pick a step size \(h\), and look at \(x\) and \(x-h\). Observe the line connecting these points. What do we see? A tiny rotation in the secant line as we move along. The distance along the curve slightly changes — the secant line shifts.

Now here’s the magic: what if \(h\) gets really, really small, almost zero? Like \(p+h \approx p\).

Whoa. The secant line turns into the tangent at that point! That’s the derivative, the instantaneous rate of change — the velocity of the function at a single point.

Mathematically, it looks like this:

\[f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\]

This is not just a formula — it’s a way to see the behavior of a function at a single instant, a peek into the instantaneous “motion” of the function. Isn’t that fascinating? Such a beauty.

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Slope

Let’s touch on slope, because that’s the feeling of change:

• Positive slope (+1) → function is going up
• Negative slope (-1) → function is going down
• Zero slope (0) → function is flat

This is exactly the idea we carry over to robotics: instantaneous joint angles, instantaneous velocities. You take a tiny moment, look at how it changes, and suddenly you can see the path the system wants to take.

Personally, I was just happy to see this concept click. To witness this phenomenon is amazing — what a beautiful creation. I feel grateful for the chance to experience this.