Every Limit Law Carries a Hypothesis

You'll learnto take a limit piece by piece when the sum, product, constant-multiple and quotient laws apply, to recognise the two ways their hypotheses fail, and to classify what direct substitution hands back before reporting it as an answer.

The limit laws let a limit be taken piece by piece — over sums, differences, products, constant multiples, powers, roots and quotients. Every one of them carries a hypothesis, and the hypothesis is stated on the pieces: each piece must have a limit of its own, and for a quotient the bottom limit must not be zero. Applying a law whose hypothesis has failed is the first way this topic goes wrong. The second is substituting and reporting whatever came back without asking which of three things it was.

8 STEPS · 6 QUICK CHECKS · EVERY LAW CARRIES A HYPOTHESIS · v1

a law applies only where its hypothesis holds ⇒ substitute, then classify what came back

Before you start

What you're looking at

Three curves at once. Two of them are the pieces — one amber, one violet — and the third, in pink, is what you get by combining them: their sum on some steps, their quotient on others.

The question

When may the limit of the combination be worked out from the limits of the pieces, and what happens on the steps where it may not?

Watch for

A shaded band closing in on the input 2 from both sides. The thickened arc inside it is the stretch of each curve still under discussion, and the heights those arcs settle on are the numbers every law in this topic is written about.

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