Relationship between l'Hospital's rule and the least upper bound property.

Question

Statement of L'Hospital's Rule

Let $F$ be an ordered field.

L'Hospital's Rule. Let $f$ and $g$ be $F$-valued functions defined on an open interval $I$ in $F$. Let $c$ be an endpoint of $I$. Note $c$ may be a finite number or one of the symbols $-\infty$,$+\infty$. Suppose $f'(x)$ and $g'(x)$ are defined everywhere on $I$. Suppose $g(x)$ and $g'(x)$ are never zero and never change sign on $I.$ Suppose one of the following two hypothesis is satisfied:

1. $\displaystyle \lim_{x \rightarrow c} f(x) = \lim_{x \rightarrow c} g(x) = 0$
2. $\displaystyle \lim_{x \rightarrow c} g(x) = \pm \infty$

Suppose $$\displaystyle \lim_{x\rightarrow c} \frac{f'(x)}{g'(x)} = L$$ where $L$ is a finite number or one of the symbols $-\infty$, $+\infty$.

Then $$\displaystyle \lim_{x\rightarrow c} \frac{f(x)}{g(x)} = L.$$

Motivation

The standard proof of l'Hospital's rule (when $F=\mathbb{R}$) uses Cauchy's mean value theorem. See:

• Taylor, A. E. (1952), "L'Hospital's rule", Amer. Math. Monthly, 59: 20–24
• https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule#General_proof
• Apostol, "Mathematical Analysis"

For an ordered field, the mean value theorem is equivalent to the least upper bound property. See:

• Equivalence of Rolle's theorem, the mean value theorem, and the least upper bound property?)
• https://arxiv.org/abs/1204.4483

Another proof of l'Hospital's rule can be based on the convergence of bounded monotone sequences and the statement that $f' > 0$ on an interval $I$ implies $f$ strictly increasing on $I$. See Taylor, A. E. (1952), "L'Hospital's rule", Amer. Math. Monthly, 59: 20–24.

The convergence of bounded monotone sequences is equivalent to the least upper bound property. The statement that $f' > 0$ on $I$ implies $f$ strictly increasing on $I$ is also equivalent to the least upper bound property.

L'Hospital's rule is false for $F=\mathbb{Q}$. The following proof outline is adapted from Exercise 7.8 of Korner's book "A Companion to Analysis." Choose a sequence $a_n \in \mathbb{R} \setminus \mathbb{Q}$ with $4^{-n-1} < a_n < 4^{-n}$ for $n=1,2,\ldots$. Define $I_0 = \{x \in \mathbb{Q} : a_0 < x \}$ and $I_n = \{x \in \mathbb{Q} : a_n < x < a_{n-1}\}$. Notice $4^{-n-1} < x< 4^{-n}$ whenever $x \in I_n$. Define $f:\mathbb{Q} \rightarrow \mathbb{Q}$ by $f(0)=0$ and $f(x) = 8^{-n}$ if $|x| \in I_n$. Remember we work in the ordered field $F=\mathbb{Q}$. We have $f'(x)=0$ for all $x \in \mathbb{Q}$, and $$ \lim_{x \to 0}\frac{f(x)}{x^2} = \infty \quad \text{and} \quad \lim_{x \to 0}\frac{f'(x)}{2x} = 0. $$

Question

What is the logical relationship between l'Hospital's rule and the least upper bound property?

Does l'Hospital's rule imply the least upper bound property? Or is there an ordered field with l'Hospital's rule but not the least upper bound property?

Related

Here is a related question. Limit of the derivative and LUB

It asks (in my notation) whether the following differentiability criteria implies the least upper bound property.

Differentiability Criteria. Let $f$ be an $F$-valued function defined on open interval $I$. Suppose $f$ is continuous on $I$. Suppose $f$ is differentiable on $I$ except at one point $c$ in $I$. If $\lim_{x \rightarrow c} f'(x)$ exists, then $f'(c)$ exists and equals this limit.

L'Hospital's rule implies this differentiability criteria. See for example https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule#Corollary

Answer 1

Here let's assume $F$ is non-archimedean.

For each infinitesimal* $x \in F^{\times}$, let $[x]$ denote the archimedean class $\bigcup \limits_{n \in \mathbb{N}^*} \left]-n |x|;-\frac{1}{n} |x|\right[ \cup \left]\frac{1}{n}\cdot|x|;n\cdot|x|\right[$ of $x$ in $F^{\times}$. Let $C$ denote the set of acrhimedean classes in $F^{\times}$.

($C$ is at least countable since for each infinitesimal $x$, $[x];[x^2];\ldots;[x^n];\ldots$ are distinct.)

Now let $\varphi$ be a choice function on $C$, and let $f$ be the map defined by $f(x) = \varphi([x])$ if $x$ is infinitesimal, and $f(y) = 0$ if $y$ is not an infinitesimal.

$f$ is differentiable on $F^{\times}$ with $f' = 0$ since for $x \neq 0$, $f$ is constant in the neighborhood $[x]$ of $x$. Moreover, $\lim \limits_{x \to 0} f(x) = 0$, since for $\varepsilon > 0$ infinitesimal, $\forall x \in \left]-\varepsilon^2;\varepsilon^2\right[$, $|f(x)| \leq \varepsilon$.

However, $\frac{f}{\operatorname{id}_{F^{\times}}}$ does not converge at $0$: given $r > 0$ and infinitesimal, and $n \in \mathbb{N}^*$, $[\frac{\varphi(r)}{n}] = [\varphi(r)]$, so $\dfrac{f(\frac{\varphi([r])}{n})}{\frac{\varphi([r])}{n}} = \dfrac{\varphi([r])}{\frac{\varphi([r])}{n}} = n$.

I believe this counterexample (inspired by that which you proposed for $\mathbb{Q}$), along with an adaptation of your argument for $\mathbb{Q}$ to any archimedean proper subfield of $\mathbb{R}$, proves that l'Hospital's rule is equivalent to the LUB property. (Incidently, the counterexample also fits my question since $f$ is not differentiable at $0$.)

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*$x$ such that $\forall n \in \mathbb{N}, n |x| < 1$.