A Limit That Fails, and a Wall That Is Not Always There

You'll learnto read the infinite notation as a description of how a limit fails rather than as a value it takes, to get the direction of each side separately from the sign of the bottom, and to decide whether a denominator zero is a wall or a hole before claiming either.

Writing that a limit is infinite looks like an answer and is really a report of a failure: the outputs pass every bound instead of settling on anything, so there is no number for the notation to name. Two things go wrong from there. The statement gets read as an existence claim, and the direction gets given once when the two sides run opposite ways. Underneath both sits a third habit, which is reading every zero of a denominator as a wall — when a factor that cancels leaves a hole instead, and only factoring says which one you have.

8 STEPS · 6 QUICK CHECKS · UNBOUNDED IS A WAY OF FAILING, NOT A VALUE · v1

a surviving factor is a wall · a cancelling factor is a hole · unbounded is not a value

Before you start

What you're looking at

One graph at a time. First a reciprocal with a dashed rule at the input where its bottom vanishes, then the same picture with the bottom squared, then a single rational expression whose bottom vanishes in two places that behave completely differently.

The question

Where do the outputs go on each side — and is a zero in the bottom by itself enough to put a wall there?

Watch for

A dashed vertical rule marks a wall. A hollow ring marks a hole: a single point the graph is missing, with the curve carrying on either side of it. Amber walks in from the left and violet from the right, and where they end up is the whole answer.

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