Four Views of One Limit, and What Each One Can Prove

You'll learnto move a limit between a graph, a table, a formula and a sentence without changing what is being claimed, to say which view is authoritative for which kind of question, and to treat a disagreement between two views as the fastest error check in the unit rather than as a paradox.

A limit can arrive as a picture, a table, a formula or a sentence, and all four describe the same object, so a correct answer has to survive translation into every one of them. The redundancy is the tool: when two views disagree, one of your readings is wrong, and finding which is faster than redoing the work. This animation puts the graph and the table on screen at once, over one shared horizontal axis, so that the commonest disagreement in the unit — a table confidently reporting a limit the graph says is not there — becomes something you can see instead of something you have to be told.

8 STEPS · 6 QUICK CHECKS · TWO VIEWS OF ONE OBJECT · v1

one function, two panels, one shared axis ⇒ when the views disagree, one of your readings is wrong

Before you start

What you're looking at

Two panels, one above the other, over the same horizontal axis. The upper panel is the graph. The lower panel is the table: it shows only the inputs the table actually lists, as separate dots, with nothing drawn between them.

The question

Do the two panels agree about where the outputs are heading as the inputs close in on 2 — and when they do not, which one is telling you the truth?

Watch for

Which side of the target each dot in the lower panel sits on. Two steps here hold the lower panel completely still and change the graph above it, and the whole answer changes with it.

Step 1 / 8