Number Strings

A string is a chain of related problems, served one at a time: the value of $\frac{x^2-9}{x-3}$ at $x=4$, then at $x=3.1$, then at $x=3.01$, then at $x=3$. Each answer is a stepping stone to the next — and in this course, the strings are quietly rebuilding calculus from scratch.

How we play

1. One problem at a time. Solve it in your head; thumb when ready.
2. A few people defend their methods; each goes on the board.
3. Before computing the next from scratch, ask: what can I reuse?

The string, walked

• “At $x=4$: top is 7, bottom is 1. Seven.”
• “At $x=3.1$: I noticed the top factors — it is $(x-3)(x+3)$, so away from 3 the whole thing is just $x+3$. That makes this one $6.1$, no long division required.”
• “At $x=3.01$: $6.01$. I can feel where this string is going.”
• “At $x=3$: nothing. Zero over zero — the one input the shortcut is not allowed to touch, because the cancelled factor is zero exactly there. The function has no value at 3. But the string just showed its destination: 6.1, 6.01, 6.001 — the outputs are converging on 6 from an input we can never use.”

Nobody was taught a definition that day. The string cornered it, and The Limit just wrote down what the room had proven: a function can have an unmistakable destination at a point where it has no value.

One variation

Run a string on slopes instead: the derivative of $x^2$ is $2x$ — of $x^3$ is $3x^2$ — of $x^5$ is $5x^4$ — so what is the derivative of $x^{100}$? Every thumb in the room goes up. Then the sting in the tail: what about $\sqrt{x}$, which is $x^{1/2}$? If the pattern holds, the exponent drops out front and steps down by one: $\frac{1}{2}{x}^{-1/2}$ — a root became a half, and the conjecture just claimed territory nobody tested it on. Whether it is entitled to that territory is the question Derivative Rules settles.

Lazy, in the best way

Mathematicians refuse to compute what they can deduce. If a problem feels brand new, look back along the string — it rarely is.