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L'Hôpital's conclusion does not imply convergence of the derivative quotient
Statement refuted
The converse of L'Hôpital's rule: if has a limit in a zero-over-zero situation, then must have a limit.
Facts & Assumptions
Given: A bounded differentiable periodic oscillator and , , for .
The derivative rules are The chain rule, in one line from Carathéodory: if is differentiable at and is differentiable at , then is differentiable at with , Sums, scalar multiples, products and quotients: , , , and when , and For a natural the function is differentiable everywhere with derivative ; for it is the constant , with derivative ; for a natural the function is differentiable at every with derivative ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term.
L'Hôpital's rule for the form at finite or infinite, one-sided endpoints asserts only the forward implication.
Counterexample
Both and tend to , and because is bounded.
Yet , which has no limit because has separated recurring values.
The quotient limit exists while the derivative-quotient limit does not, so the converse fails.
Depends on
- A bounded $C^1$ periodic oscillator made from a quartic Hermite spline
- L'Hôpital's rule for the $0/0$ form at finite or infinite, one-sided endpoints
- The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f \circ g$ is differentiable at $c$ with $(f \circ g)'(c) = f'(g(c))\,g'(c)$
- Sums, scalar multiples, products and quotients: $(f+g)'(c) = f'(c) + g'(c)$, $(\alpha f)'(c) = \alpha f'(c)$, $(fg)'(c) = f'(c)g(c) + f(c)g'(c)$, and $(f/g)'(c) = \bigl(f'(c)g(c) - f(c)g'(c)\bigr)/g(c)^{2}$ when $g(c) \ne 0$
- For a natural $n \ge 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $\iota(n)\,x^{\,n-1}$; for $n = 0$ it is the constant $1$, with derivative $0$; for a natural $n \ge 1$ the function $x \mapsto x^{-n}$ is differentiable at every $x \ne 0$ with derivative $-\iota(n)\,x^{-n-1}$; consequently every polynomial function is differentiable at every real, with the derivative computed term by term
Used by
Nothing in the library uses this result yet.
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 75 results over 20 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 lecture notes (standard reference, not scraped)