Choosing a Limit Procedure

You'll learnto open every limit with substitution, to classify what comes back as a determinate value, an indeterminate form or a vanishing denominator, and to let that classification — rather than the look of the expression — choose the rewrite.

You now have every technique in the unit, so the work is no longer doing one — it is choosing one. The routine is fixed and it is one line long: substitute, then read what came back. A determinate number is the answer and you stop. A zero over a zero is a signal that more work is needed, and the structure of the expression names which work. A nonzero over a zero means the outputs run away, which is a verdict rather than a dead end. Reaching for factoring before you know the form costs time; reporting the form as an answer costs marks.

8 STEPS · 6 QUICK CHECKS · SUBSTITUTE FIRST, THEN READ THE FORM · v1

substitute first · a number stops · 0/0 rewrites · k/0 runs away

Before you start

What you're looking at

One graph with one target marked at x = 2, and three different functions drawn on it in turn. The axes, the scale and the target never move, so the only thing that changes between the pictures is what happens at the target.

The question

Substitution is always the first move. Which of the three things it can return is this, and what does that return value tell you to do next?

Watch for

A solid disc means the value was there all along. A hollow ring means the point is missing and a rewrite will find it. Two branches leaving the top of the frame with no ring at all mean there is no height to name.

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