Rewriting an Expression Without Changing Its Limit

You'll learnto read a zero over zero as an instruction rather than an answer, to resolve it by factoring, by a conjugate or by a common denominator, and to tell a cancellation that removes one point from the picture apart from one that replaces the whole curve.

A zero over zero is not a value and not a verdict. It says that the top and the bottom both vanish at the target, which means both carry the same factor, which means the expression can be rewritten into one that agrees with it at every nearby input. That is all a limit ever consults, so the rewritten form answers the original question. The whole technique turns on one distinction: a factor divides the entire numerator, while a term is only part of a sum, and striking a term does not simplify an expression — it swaps it for a different one.

8 STEPS · 6 QUICK CHECKS · A FACTOR, NEVER A TERM · v1

a zero over zero is an instruction · cancel a factor of the whole side, never a term of a sum

Before you start

What you're looking at

One graph. The pink stroke is always the expression as it was written; a dashed emerald stroke is what a rewrite produces, and a red stroke is what an invalid one produces.

The question

When substitution returns zero over zero, what may you do to the expression — and how would you know if you had done something that changed it?

Watch for

A valid cancellation leaves the stroke exactly where it was and takes away a single point, drawn as a hollow ring. An invalid one puts a whole new stroke somewhere else. Step 6 draws both at once.

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