Limits

The Big Drop’s display should show the rider’s speed at each moment. Your average-speed code from lesson 1 measures the drop between $t = 1$ and $t = 1 + k$:

To get the speed at $t = 1$, you’d set the gap $k$ to 0. But then the top is $h(1) - h(1) = 0$ and the bottom is 0: the program crashes with a division by zero. The question isn’t silly, though. As the gap gets smaller and smaller, the average speed clearly settles on something.

Ancient Greek philosophers argued about exactly this. Zeno said that at any single instant a flying arrow covers no distance, so how can it be moving? It took until the 1800s for mathematicians (Cauchy, then Weierstrass) to give a precise answer: don’t plug in zero, ask what the values approach. That’s a limit.

The real words

• The Limit:

  $$\lim_{x \to c} f(x) = L$$

  means you can make $f(x)$ as close to $L$ as you like by taking $x$ close enough to $c$, with $x \neq c$. The precise “epsilon–delta” version of this sentence was written down by Weierstrass in the 1800s.

• What happens at $c$ is irrelevant: $q(0)$ is undefined, yet $\lim_{k \to 0} q(k) = -9.8$. Algebra shows why: $h(1 + k) - h(1) = -4.9(2k + k^2)$, so $q(k) = -9.8 - 4.9k$ for every $k \neq 0$, which heads to $-9.8$.

• One-sided limits: $\lim_{x \to c^-}$ uses only $x < c$; $\lim_{x \to c^+}$ uses only $x > c$. The two-sided limit exists exactly when both exist and are equal.

• How limits fail:

  • a jump: the sides approach different values;
  • blowing up: $1/x$ near 0 grows without bound (we write $\to \infty$, but there’s no finite limit);
  • wild wiggling: $\sin(1/x)$ swings between −1 and 1 infinitely often near 0 and never settles.

• A function is continuous at $c$ when there’s no surprise: $\lim_{x \to c} f(x) = f(c)$. Polynomials, $\sin$, $\cos$ and $e^x$ are continuous everywhere, so for them you can just plug in.

• Limits at infinity: the pool’s volume $450(1 - e^{-t/4})$ approaches 450 as $t \to \infty$ without ever quite reaching it.