A Rule You Have to Earn the Right to Use

You'll learnto substitute before you differentiate, so the rule is only ever applied to a form that qualifies — and to read the number it returns as a statement about the original quotient, not about a function value or a derivative.

Units 1 and 2 left some limits unfinished. L’Hospital’s Rule finishes them, and it is the most over-applied result in the course. The rule itself is one line: replace the limit of f/g with the limit of f′/g′. The precondition is the part that matters. Substitution has to come back 0/0 or ∞/∞ first, and if it comes back 0/8 or 5/0 instead, differentiating top and bottom still produces a number — a clean, confident, ordinary-looking number that is not the limit. Nothing in that wrong answer looks wrong. Only the skipped check catches it.

8 STEPS · 6 QUICK CHECKS · SUBSTITUTE BEFORE YOU DIFFERENTIATE · ONLY 0/0 AND ∞/∞ OPEN THE GATE · v1

substitute → read the form → only 0/0 and ∞/∞ open the gate → then f′/g′

Before you start

What you're looking at

A gate. The bar across the top substitutes x = c into the numerator and the denominator, prints the form it gets back, and then either opens or stays shut. The panel underneath plots the quotient the gate is reading.

The question

Is this limit even one the rule is allowed to touch — and if the rule returns a number anyway, what is that number actually about?

Watch for

The two limits that share a numerator and differ by one sign in the denominator. The rule hands back 3 for both. It is the answer for one of them and a fabrication for the other.

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