Reading a Limit Off a Table, and Knowing What It Proves

You'll learnto build a table that closes in on a target from both sides, to read the estimate off where the digits agree rather than off the last row, to treat an indeterminate form as a prompt for more work rather than as an answer, and to say what a finite table can and cannot establish.

A table is the most direct way to watch a limit happen: choose inputs marching toward the target, read the outputs, and see what they settle on. It is also the easiest place to fool yourself, because a table that stopped too early and a table that has settled look identical on the page. Everything in this topic is about the difference between the two, and about what the row you can never fill is telling you.

8 STEPS · 6 QUICK CHECKS · SAMPLE BOTH SIDES AND GO CLOSE · v1

two columns march in from opposite sides ⇒ read where they agree, never the row at the target

Before you start

What you're looking at

One graph and one table, side by side, showing the same thing twice. Two dots walk in along the curve toward a target on the horizontal axis, and the table fills in a row for every place they stop.

The question

What value are the two columns settling on — and how would you know the difference between a column that has settled and one that merely has not moved much yet?

Watch for

The middle row of the table is never filled. Watch also for the moment the two dots become one on screen while the table is still reading two different rows: that is the point at which the picture stops being the better instrument.

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