Calculus I

Getting Arbitrarily Close

Or: What a function is doing right next to a point it might never reach

Companion notebook. Play with this chapter’s ideas — sliders, animations, and drills — in notebooks/ch2_limits.ipynb.

By the end of this chapter you will be able to find what a function approaches as its input slides toward a target — which is the same idea as Chapter 1, with one new and crucial wrinkle. You will compute these limits three ways: by simply plugging in (when the function is well-behaved), by factoring and canceling (when plugging in gives the nonsense $\tfrac00$), and out at infinity. You will know what it means for a function to be continuous — the “draw it without lifting your pen” idea — and why, underneath, continuity is really the promise that a small change in input makes only a small change in output. You’ll also meet the two great theorems that continuity guarantees — the Intermediate Value Theorem (a continuous function can’t skip a value, which is how we’ll prove that solutions exist) and the Extreme Value Theorem (a continuous function on a closed interval actually reaches its highest and lowest points) — both of which the back half of the book quietly runs on.

And you will meet one specific limit, $\dfrac{\sin x}{x}\to 1$, that looks like it should be undefined and isn’t. It is the single most important limit in the book, because in the very next chapter it turns out to be a slope — your first derivative, computed before we even have the word.

Here is the one new idea, and it is worth saying before anything else. In Chapter 1 the input was a whole number $n$ that marched off to infinity. Now the input is a real number $x$ that you slide toward a finite point $a$ — and it can approach $a$ from either side. The limit asks what $f(x)$ is doing in the neighborhood of $a$. It does not ask, and does not care, what $f$ does at $a$ itself. That gap between “near $a$” and “at $a$” looks like a technicality. It is actually the loophole that makes all of calculus possible.

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