Three Ways a Graph Can Break

You'll learnto name a discontinuity from its two one-sided limits rather than from how the formula looks, to tell a denominator factor that cancels from one that survives, and to say which of the three types can be repaired at all.

There are exactly three ways a function can fail to be continuous at a point, and the two one-sided limits decide which one you are looking at. The algebra's appearance decides nothing: a zero in the denominator can produce a hole or a vertical asymptote, and only factoring says which. One expression can carry one of each, at two different places, which is why the shortcut that reads every denominator zero as a blow-up is wrong about half of them.

8 STEPS · 6 QUICK CHECKS · THE ONE-SIDED LIMITS NAME THE TYPE · v1

the two one-sided limits name the break, never the look of the formula

Before you start

What you're looking at

One graph carrying three different breaks at once: a hole, a jump, and a vertical asymptote. Later steps switch to a single rational expression whose denominator vanishes at two places, one of which is a hole and the other an asymptote.

The question

For each break, what are the two one-sided limits doing — and is that the same question as what the formula looks like?

Watch for

A hollow ring marks a height the outputs approach. A solid dot marks a value the function actually takes. Amber walks in from the left, violet from the right, and it is what those two colours arrive at that names the type.

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