What a Limit Says, and What It Refuses to Say

You'll learnto read the limit notation as a claim about the neighbourhood of a point rather than about the point, to take the two one-sided limits separately and compare them, and to say that a limit does not exist without inventing a value for it.

The limit of f at a is a statement about inputs near a, with the input a itself deliberately left out. That exclusion looks like a technicality and is the whole point: it is what lets a limit exist where the function does not, and what lets a limit disagree with a value that is sitting right there on the graph. Everything expensive in this topic happens when the two directions of approach behave differently, because then there is nothing for the notation to name.

8 STEPS · 6 QUICK CHECKS · BOTH SIDES MUST EXIST AND AGREE · v1

the outputs near a, never the output at a ⇒ both sides must exist and agree

Before you start

What you're looking at

One graph, with a target value marked on the horizontal axis. Two dots walk in along the curve toward that target, one from the left and one from the right, and their heights are listed as they close in.

The question

Where are the outputs heading as the inputs close in on the target — and is that the same question as what the function does at the target?

Watch for

A hollow ring marks the height the curve approaches. A solid dot marks the value the function actually takes. They are different objects, and several steps here move one while holding the other still.

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