Repairing a Hole, and the One Value That Does It

You'll learnto decide from the two one-sided limits whether a break can be repaired at all, to take the repair value from the limit rather than from the expression that failed, and to tell the algebra that computes that value from the definition that actually uses it.

A hole is the one break that can be patched, and exactly one number patches it. That number is the limit, and it is not what you get by substituting into the expression that broke — substituting is what produced the break. Cancelling a common factor is worth doing and it repairs nothing on its own: it computes the number, and the repair is the separate act of writing a new function that takes that number at the point. Everything else on this page follows from those two sentences.

8 STEPS · 6 QUICK CHECKS · THE REPAIR VALUE IS THE LIMIT · v1

the repair value is the limit, and cancelling only computes it

Before you start

What you're looking at

One rational curve carrying two breaks at once: a hole at x = 1, where a factor cancels, and a vertical wall at x = 4, where one does not. Step 7 switches to a second curve with a jump in it.

The question

Which break can be repaired, what exactly do you assign at the point — and is writing the cancelled expression the same act as assigning it?

Watch for

A hollow ring marks a height the outputs approach with nothing plotted there. A filled point marks a value the function actually takes. The repair is the ring becoming a point, and it only works at one height.

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