Limits and Derivatives

Published 01/27/2021

Limits are all about filling in holes in graphs. Derivatives are all about finding the slope at individual points on a graph. It can be hard to see how these two concepts relate (besides the fact that they both have to do with graphs).

But it makes more sense if you think about the formula for slope as a 3D graph \(g\):

$$ \begin{align} f(x) &= \sin(x)\\ g(a, b) &= \frac{f(a) - f(b)}{a-b}\\ &= \frac{sin(a) - sin(b)}{a - b}\\ \end{align} $$

The formula for slope has two inputs (the first point and the second point) and one output (the slope between these two points). This formula creates a bunch of holes along the diagonal. Every place where the two inputs are the same has an undefined slope. This is another way of saying that we can’t find the slope at a point.

But if we rearrange the slope formula so that its inputs are (1) a point, and (2) a distance to the next point, we can easily apply our standard limit rules to this, so that we can fill in those holes.

$$ \begin{align} g(a, h) &= \frac{f(a) - f(a + h)}{a-(a + h)}\\ &= \frac{sin(a) - sin(a + h)}{h}\\ f’(a) &= \lim_{h\to 0} \frac{sin(a) - sin(a + h)}{h}\\ \end{align} $$

So basically, taking a derivative is just about using limits to fill in the holes in this very special kind of 3D graph.