The bookUnit 1 · Continuity and Differentiation6 / 83

Differentiability, mean value theorems, Taylor, L'Hôpital

Why this is asked: MVT-family questions are about *which hypothesis is missing*: Rolle needs continuity at the endpoints, Darboux needs only differentiability, L'Hôpital needs the derivative-quotient limit to exist. Know x² sin(1/x) cold.

Differentiation — MVT, Darboux, Taylor and the classic spoilers

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Where L'Hôpital's rule quietly failsinteractive

A limit that exists but which L'Hôpital never finds — and the hypothesis everyone forgets to check.

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Check yourself — select all that apply

Let $f : \mathbb{R} \to \mathbb{R}$ be differentiable. Which of the following are true?

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