Chapter 8Limits and the definition of derivatives

"Calculus is all about making curvy things look straight" Anonymous Professor

The last several lessons have been about the idea of a derivative, and before moving on to integrals, I want to take some time to talk about limits.

To be honest, the idea of a limit is not really anything new. If you know what the word "approach" means you pretty much already know what a limit is, you could say the rest is a matter of assigning fancy notation to the intuitive idea of one value that just gets closer to another. But there are a number of reasons to dive deeper into this topic.

For one thing it’s worth showing is how the way I’ve been describing derivatives so far lines up with the the formal definition of a derivative as it’s typically presented in most courses and textbooks. I want to give you a little confidence that thinking of terms like dxdx and dfdf as concrete non-zero nudges is not just some trick for building intuition; it’s actually backed up by the formal definition of a derivative in all of its rigor.

Formal Definition of the derivative

Let’s take a look at the formal definition of the derivative. As a reminder, when you have some function f(x)f(x), to think about the derivative at a particular input, maybe x=2x=2, you start by imagining nudging that input by some tiny dxdx, and looking at the resulting change to the output, dfdf.

The ratio df/dxdf/dx, which can nicely be thought of as the rise-over-run slope between the starting point on the graph and the nudged point, is almost what the derivative is. The actual derivative is whatever this ratio approaches as dxdx approaches 00.

Just to spell out what is meant here, that nudge to the output "dfdf" is is the difference between ff at the starting-input plus dxdx and ff at the starting-input; the change to the output caused by the nudge dxdx.

To express that you want to find what this ratio approaches as dxdx approaches 00, you write "lim", for limit, with "dxdx arrow 00" below it.

Now, you’ll almost never see terms with a lowercase dd, like dxdx, inside a limit like this. Instead the standard is to use a different variable, like Δx\Delta x (delta xx), or commonly "hh" for some reason.

The way I like to think of it is that terms with this lowercase dd in the typical derivative expression have built into them the idea of a limit, the idea that dxdx is supposed to eventually approach 00.

So in a sense this lefthand side "df/dxdf/dx", the ratio we’ve been thinking about in the past lessons, is just shorthand for what the righthand side spells out in more detail, writing out exactly what we mean by dfdf, and writing out this limit process explicitly. And that righthand side is the formal definition of a derivative, as you’d commonly see it in any calculus textbook.

The formal derivative definition. Notice that nothing about this definition references the idea of an "infinitely small" change.

No infinitely small rant

Now, if you’ll pardon me for a small rant here, I want to emphasize that nothing about this righthand side references the paradoxical idea of an "infinitely small" change. The point of limits is to avoid that.

This value hh is the exact same thing as the "dxdx" I’ve been referencing throughout the series. It’s a nudge to the input of ff with some non-zero, finitely small size, like 0.0010.001, that we’re analyzing for arbitrarily small choices of hh. In fact, the reason people introduce a new variable name into this formal definition, rather than just using dxdx, is to be clear that these changes to the input are ordinary numbers that have nothing to do with infinitesimals.

Rather than interpret dxdx as an "infinitely small change", as others like to do, I think you can and should interpret dxdx as a concrete, finitely small nudge, just so long as you remember to ask what happens as it approaches 00. This method helps build a stronger intuition for where the rules of calculus come from and, more than that, it’s not just some trick for building intuitions. Everything I’ve been saying about derivatives with this concrete-finitely-small-nudge philosophy is a translation of the formal definition of derivatives.

Long story short, the big fuss about limits is that they let us avoid talking about infinitely small changes by instead asking what happens as the size of some change to our variable approaches 00.

Why we care

So why do we teach the limit notation and how does it help us formalize what it means for one value to approach another?

Maybe you are convinced that, on an intuitive level, the language and tools we have invented so far give us what we need to solve these types of problems. When thinking about small nudges to a function's input, as long as we imagine that nudge approaching zero, things usually work out. In the same way, we can zoom in on a point on a curve and convince ourselves that if we zoom in far enough the curve really starts to look straight.

Zoom in on the exponential function exe^x at x=1x=1 and the rate of change (slope) starts to look a whole lot like 11.

But, you may still have doubts and questions, as certainly some of the creators of calculus did. For example, just how far do we have to zoom in on a point before a function starts to look like its tangent line at that point?

If you are applying calculus, your primary concern is probably the accuracy of the approximation and you might play a cat and mouse game to choose a small enough dxdx so that the approximation is is accurate enough to calculate a useful result. A mathematician, on the other hand, is playing a different game; they are interested in whether they can rig the cat and mouse game of choosing and smaller values and getting a more accurate result so that they will always win.

Even more than this, mathematicians care about ensuring all their statements have a precise and airtight meaning, which does not rest on visual intuition. As sensible as it might seem to say that this ratio approaches a certain value, and to draw a line through the graph whose slope approaches that of a tangent line, those seeking rigor insist on something better. In the next chapter, we’ll take a look the epsilon-delta definition of limits, which finally puts this idea of "approaching" onto a firm foundation.

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